# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 4 Damping of the Building Model
### Coupled Equations of Motion  
As presented in Sections 4.4 and 4.5, the damping and mitigation performance of the viscous inerter damper is assessed by root locus and frequency response analyses. For that purpose, the present section introduces the coupled equations for the damped structure in Figure 1. The equations also include the stiffness \(k_{0}\) in Figure 7, which vanishes (\(k_{0}=0\)) in the ideal viscous inerter damper but is included in the numerical example to avoid drift between inerter and dashpot.  
A complex frequency--or pole--is derived from the damped eigenvalue problem, governed by the coupled equations (5) for the structure and (26) for the viscous inerter damper. It is convenient to eliminate the force \(f=-\omega^{2}y\), with \(y=u-v\) being the deformation across the inerter. Thereby, the governing Equations (5) and (26) can be written in coupled form as  
\[\left(-\omega^{2}\bar{\mathbf{M}}+i\omega\bar{\mathbf{C}}+\bar{\mathbf{K}} \right)\bar{\mathbf{q}}=\bar{\mathbf{f}} \tag{45}\]  
introducing the augmented matrices  
\[\bar{\mathbf{M}}=\left[\begin{array}{cc}\mathbf{M}&\mathbf{0}\\ \mathbf{0}^{T}&m\end{array}\right]\quad,\quad\bar{\mathbf{C}}=\left[\begin{array} []{cc}\mathbf{C}+c\mathbf{b}\mathbf{b}^{T}&-c\mathbf{b}\\ -c\mathbf{b}^{T}&c\end{array}\right]\quad,\quad\bar{\mathbf{K}}=\left[ \begin{array}{cc}\mathbf{K}+k_{0}\mathbf{b}\mathbf{b}^{T}&-k_{0}\mathbf{b}\\ -k_{0}\mathbf{b}^{T}&k_{0}\end{array}\right] \tag{46}\]  
for the augmented displacement and load vector  
\[\bar{\mathbf{q}}=\left[\begin{array}{cc}\mathbf{q}\\ y\end{array}\right]\quad,\quad\bar{\mathbf{f}}=\left[\begin{array}{cc}\mathbf{ f}_{ext}\\ 0\end{array}\right] \tag{47}\]  
Figure 7: Inerter damper with front-end inertance \(m\), in series with visco-elastic dashpot: \(D(\omega)=k_{0}+i\omega c\).  
where the latter assumes that the inerter damper is unexposed to external forcing. These augmented matrices correspond to those derived for a classic TMD, with \(y\) representing the motion of the auxiliary TMD mass.  
The complex system poles (or natural frequencies) are governed by the coupled equation of motion (45) without external loading (\(\mathbf{\tilde{f}}=\mathbf{0}\)). This eigenvalue problem is conveniently solved in the first-order state space format  
\[(\mathbf{A}-i\omega\mathbf{I})\mathbf{z}=\mathbf{0} \tag{48}\]  
where the system matrix and state-space vector are introduced as  
\[\mathbf{A}=\left[\begin{array}{cc}\mathbf{0}&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array}\right]\quad \quad,\quad\quad\mathbf{z}=\left[\begin{array}{c}\mathbf{\tilde{q}}\\ i\omega\mathbf{\tilde{q}}\end{array}\right] \tag{49}\]  
It is noted that in (45) to (49), the entry \(\mathbf{0}\) is the zero vector or matrix and \(\mathbf{I}\) is the identity matrix that fit the given dimensions.  
In Section 4.4, the first-order eigenvalue problem (48) is solved for a varying viscous coefficient \(c\) to trace the complex roots in a root locus diagram, as already applied in Figure 6. In Section 4.5, the frequency response curves are obtained by solving the frequency-domain equation of motion (45) for optimal tuning and varying (real-valued) angular frequency \(\omega\). These analyses will illustrate the ability of the viscous inerter damper to realize apparent resonant damping, with \(k_{0}\) being sufficiently small to not interfere while still large enough to avoid any drift across the viscous dashpot.