# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 5 Conclusions  
In the tuning and design of inerter-based vibration absorbers, the inclusion of correction for residual modes is important, as inerter dampers have the ability to impose sufficiently large changes in the expansion mode shape(s) to influence the tuning accuracy. In the present case, the very simple viscous inerter damper is introduced, for which the residual-mode inertia correction adjusts the effective mass ratio, while the corresponding stiffness corrections enable resonant tuning by modal interaction between the fundamental structural vibration modes. An iterative tuning procedure is derived for the optimal inertance based on the requirement of inverse clamped roots relative to the free (undamped) natural frequency. The present tuning--in its current form--directly hinges on the balanced pole placement calibration that secures equal modal damping, which means it must be altered if alternative tuning expressions or principles are to be used.  
A 20-story building model is used as an example. The root locus and frequency response analysis show that the resonant damping characteristics are almost perfectly realized by the proposed tuning method. The bifurcation point is not entirely obtained, and a small inclination--relative to the optimal case--is noted for the frequency response curves. However, the resonant tuning of the non-resonant viscous inerter damper is a novel concept that (to the author's knowledge) has not been explored before.  
For the present civil engineering application, the optimal mass ratio of \(\mu=0.1051\) is not extraordinarily large. However, it results in an extremely large inertance \(m=919.4\times 10^{6}\,\mathrm{kg}\) because the modal mass \(m_{1}=8746\times 10^{6}\,\mathrm{kg}\) for the first vibration mode is almost 100 times larger than the total translational mass of the building itself (\(20\times 5\times 10^{6}\,\mathrm{kg}=100\times 10^{6}\,\mathrm{kg}\)). Note that the modal mass in (14) is much larger than the actual building mass because the normalization of the mode shape in (10) enforces a unit deflection at the bottom floor and not at the top floor, where the deflection is otherwise largest for mode \(j=1\). However, the inerters can be constructed with enormous gearing ratios, and although the present inerter can be placed on the ground, its size seems enormous, and it may require some development to make the concept feasible. However, if it can be constructed, the large mass ratio implies large attainable damping, which may reduce the construction material needed for reinforcement against vibrational loading. Finally, the robustness of the proposed damper tuning should be investigated in further detail, especially addressing the damper's sensitivity to changes in the inertance \(m\), which for the present damper configuration must be chosen precisely to achieve the desired resonant damper characteristics.  
This research received no external funding  
**Institutional Review Board Statement:** Not applicable  
**Informed Consent Statement:** Not applicable  
**Data Availability Statement:** The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.