# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 3 Viscous Inerter Damper
### Single-Mode Model  
For the pure viscous inerter damper in Figure 4, the non-dimensional damper function \(\eta\) in (22) can be expressed as  
\[\eta=\frac{i\omega c}{k_{s}}=i\rho\beta \tag{27}\]  
which introduces the non-dimensional damper ratio  
\[\beta=\frac{c}{\sqrt{k_{s}m_{s}}} \tag{28}\]  
Note that \(c\) is normalized by the modal mass and stiffness (and not the damper mass and stiffness). The damper function \(\eta=i\rho\beta\) is then substituted into the modal frequency response function (21), which upon division of both numerator and denominator by \(i\rho\beta\) can be written as  
\[\frac{r_{s}k_{s}}{f_{s}}=\frac{\bigg{[}i\rho\frac{\bar{\mu}_{s}}{\beta}+1- \frac{\rho^{2}\bar{\mu}_{s}}{\kappa^{\prime}_{s}}\bigg{]}}{(-\rho^{2}+1) \bigg{[}i\rho\frac{\bar{\mu}_{s}}{\beta}+1-\frac{\rho^{2}\bar{\mu}_{s}}{ \kappa^{\prime}_{s}}\bigg{]}-\rho^{2}\bar{\mu}_{s}} \tag{29}\]  
with the damping parameter \(\beta\) appearing in its reciprocal form. This FRF describes a single-dof structure model with modal coordinate \(r_{s}\) and the augmented damper model in Figure 3 attached. This simplified structural model is shown in Figure 5, in which \(m^{\prime}_{s}\) (contained in \(\bar{m}_{s}\)) and \(k^{\prime}_{s}\) represent the interaction with the other modes (\(j\
eq s\)). Thus, these correction parameters introduce the modifications needed to secure accurate modal results when reducing the system from the full mdof building mode in Figure 1b to the 2-dof (modal) model in Figure 5.  
Figure 4: Inerter damper with front-end inertance \(m\) in series with viscous dashpot, represented by the damper function \(D(\omega)=i\omega c\). The correction coefficients \(m^{\prime}_{\rm s}\) (or \(\bar{m}_{s}\)) and \(k^{\prime}_{s}\) are determined in the next section so that the system poles (complex natural frequencies) approach the correct (real-valued) natural frequencies in the clamped limit. The characteristic equation, which governs the system poles, is identified from the denominator of the FRF in (21) as  
\[(-\rho^{2}+1)\bigg{(}1-\rho^{2}\frac{\bar{h}_{s}}{k^{\prime}_{s}}\bigg{)}-\rho ^{2}\bar{u}_{s}+i\rho\,\frac{\bar{\mu}_{s}}{\beta}\,(-\rho^{2}+1)=0 \tag{30}\]  
It is worth noticing that finite values of \(\kappa^{\prime}_{s}\) will imply that this equation has two complex conjugate solution pairs. For the viscous inerter damper with a small front-end inertance, the damper will act as a viscous damper, whereby the stiffness correction term must vanish for the characteristic equation to only have a single pair of complex conjugate poles. However, in the present design, the front-end inertance is increased until coupling with an adjacent vibration mode allows two system poles to interact and thereby create apparent resonant damper properties. Figure 6 shows the trajectories of the complex system roots for the full 20-dof building model when attaching a pure viscous inerter damper. The equations of motion and the corresponding eigenvalue problem that governs the complex poles are presented in detail in Section 4. The blue circles \"Fine with blue font\" on the real frequency axis represent the free natural frequency \(\rho_{j}=\omega_{j}/\omega_{1}\). Thus, for mode \(j=s=1\), the blue circle is by definition located at \(\rho=1\) when \(\eta=0\). On the other hand, the blue squares represent the clamped natural frequencies at \(\rho=\partial_{j}/\omega_{1}=\hat{\rho}_{j}\), which the complex trajectories reach when \(\eta\to\infty\). In the top subplot in Figure 6a, the front-end mass \(m=0.067m_{1}\) chosen is slightly too small, as the damper trajectory approaches its clamped limit (blue square) at around \(\rho\simeq 1.3\). For this inertance, the mode 1 locus is very small with limited attainable damping because of the very limited modal interaction between the mode 1 locus and the damper locus. In the next subplot (b), the front-end mass increases to \(m=0.10m_{1}\), which clearly implies a great interaction with the mode 1 locus.