# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 2 Governing Equations of Motion
### Limiting Eigenvalue Problems  
A typical damper design aims at optimizing the damping in a single vibration mode, which dominates the structural response of the structure. For high-rise buildings and other slender structures, such as wind turbines, towers, and masts, the dominant mode is likely the first (or fundamental) vibration mode. An effective system reduction procedure must isolate the targeted vibration mode, referred to below as mode number \(s\), while still accounting for the spillover from all the other modes. For that, the present subsection introduces the limiting eigenvalue problems associated with \(D\to 0\) and \(\infty\), from which the corresponding natural frequencies are used to calibrate the modal spill-over to mode \(s\) from the non-targeted modes.  
In Section 2.3, the modal expansion is based on the undamped vibration modes associated with \(D=0\) and thereby \(f=0\) from (6). In this free damper limit, the frequency equation of motion (5) simplifies to the generalized eigenvalue problem  
\[\Big{(}-\omega_{f}^{2}\mathbf{M}+\mathbf{K}\Big{)}\mathbf{u}_{j}=\mathbf{0} \tag{8}\]  
for vanishing damping and external loading. In (8), the angular natural frequency \(\omega_{j}\) is determined from the eigenvalue, while \(\mathbf{u}_{j}\) is the associated mode shape vector for vibration  
Figure 2: General inerter damper with front-end inertance \(m\) in series with general damper element \(D(\omega)\).  
mode \(j\). Below, this undamped limit for \(D\to 0\) is denoted as the free condition, whereby (8) is referred to as the free eigenvalue problem.  
In the other limit \(D\to\infty\)--referred to below as the clamped limit--the damper element (inside the dashed-blue box) in Figure 2 will fully lock, whereby the generalized eigenvalue problem will change into  
\[\left(-\hat{\omega}_{j}^{2}(\mathbf{M}+m\mathbf{b}\mathbf{b}^{T})+\mathbf{K} \right)\hat{\mathbf{u}}_{j}=\mathbf{0} \tag{9}\]  
where \(m\) will increase the system's inertia at the damper location. The hat symbol \(\left(\hat{\cdot}\right)\) depicts solutions or parameters associated with the clamped limit, e.g., the clamped natural frequency \(\hat{\omega}_{j}\) and clamped mode shape vector \(\hat{\mathbf{u}}_{j}\). For the target mode \(j=s\), it is assumed that the addition of inertia at the damper location will result in the clamped natural frequency being smaller than the associated free natural frequency, i.e., \(\hat{\omega}_{s}<\omega_{s}\).  
In the next section, the free mode shapes are conveniently used in the modal expansion, as these mode shapes do not depend on the front-end inertance \(m\) of the inerter damper in Figure 2. Therefore, the clamped natural frequencies are used below to determine the influence from other modes, which in the present damper is of great importance, as the resonant damper characteristics are derived from modal interaction and not by an explicit damper resonant. The derivations are presented in the most compact form when both the free and clamped mode shapes are normalized to unit damper deflections, enforced by the relations  
\[\mathbf{b}^{T}\mathbf{u}_{j}=1\qquad,\qquad\mathbf{b}^{T}\hat{\mathbf{u}}_{j}=1 \tag{10}\]  
For the indirect location of the inerter damper in Figure 1, the modal mass will be much larger than for a TMDI placed at the top floor, as in [14; 39].