# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 2 Governing Equations of Motion
### Residual Mode Correction  
The damper force relation in (6) must be modified to give the damper force \(f\) as a function of (not \(u\), but) the modal coordinate \(r_{s}\) for the target mode \(s\). This modal damper force relation will inherently contain spill-over from other modes (\(j\
eq s\)), simply because the placement of a local damper, presently on the first floor, violates the modal orthogonality conditions. Substitution of the modal damper deflection representation (12) into the damper force relation (6) gives  
\[\bigg{(}\frac{1}{-\omega^{2}m}+\frac{1}{D(\omega)}\bigg{)}f=r_{s}+\sum_{j\
eq s }r_{j}=r_{s}-\underbrace{\sum_{j\
eq s}\frac{1}{-\omega^{2}m_{j}+k_{j}}}_{1/R( \omega)}f=r_{s}-\frac{f}{R(\omega)} \tag{16}\]  
On the right-hand side of the first equality, the sum of \(r_{j}\) is separated into \(r_{s}\) for the target mode and a summation over all other (residual) modes \(j\
eq s\). The modal coordinates for these residual modes are replaced by the modal Equation (13) for \(f_{j}=0\), which must be accurate for isolated tuning to mode \(j=s\). Finally, in (16), all dynamic residual mode terms have \(f\) as a common factor, whereby they can be used in a general frequency-dependent residual mode function \(R(\omega)\), to be determined below. When collecting all terms proportional to the damper force \(f\) on the left-hand side, the inerter Equation (16) can be reduced to  
\[\bigg{(}\frac{1}{-\omega^{2}m}+\frac{1}{D(\omega)}+\frac{1}{R(\omega)}\bigg{)} f=r_{s} \tag{17}\]  
which constitutes the governing damper equation for the target mode \(j=s\). The closed-loop equations for mode \(s\) are thus composed of the modal Equation (13) for \(j=s\) and the inerter damper force Equation (17). As argued in [18, 44, 45, 46], the influence from residual modes may be split into two separate components, representing stiffness and inertia. For the residual mode function \(R(\omega)\) in (16), this separation can be expressed as  
\[\frac{1}{R(\omega)}\simeq\frac{1}{-\omega^{2}m_{s}^{\prime}}+\frac{1}{k_{s}^{ \prime}} \tag{18}\]  
with \(m_{s}^{\prime}\) and \(k_{s}^{\prime}\) representing artificial inertia and stiffness components that account for the interaction with other modes (\(j\
eq s\)). For the present inerter damper, the substitution of this \(1/R\) into the force relation (17) gives  
\[\bigg{(}\frac{1}{-\omega^{2}\hat{m}_{s}}+\frac{1}{D(\omega)}+\frac{1}{k_{s}^{ \prime}}\bigg{)}f=r_{s} \tag{19}\]  
which directly defines the combined inertance \(\hat{m}_{s}\) by the reciprocal relation  
\[\frac{1}{\hat{m}_{s}}=\frac{1}{m}+\frac{1}{m_{s}^{\prime}} \tag{20}\]In the case of vanishing influence from residual modes, the reciprocal inertia correction will vanish (\(1/m_{s}^{\prime}\to 0\)), whereby the combined inertance recovers the front-end inertance, i.e., \(\bar{m}_{s}\to m\). The modal damper model is shown in Figure 3, representing the modal damper relation (19) with combined inertance \(\bar{m}_{s}\) and residual mode stiffness \(k_{s}^{\prime}\). In the upcoming Section 3, these two system parameters are determined for a pure viscous inerter damper with a damper function \(D=i\omega c\). For target mode \(j=s\), the frequency response amplitude is conveniently derived from a frequency response function (FRF), which is obtained by eliminating \(f\) between (13) and (17).