# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 2 Governing Equations of Motion  
Figure 1a shows a simple shear frame building model with \(n=20\) stories. The story mass is assumed to be concentrated (lumped) at each floor, where it is denoted as \(m_{f}\), while \(k_{f}\) is the corresponding inter-story stiffness, as depicted in Figure 1b. The governing equation of motion in the time domain is then written in the common form as  
\[\mathbf{M}\ddot{\mathbf{q}}(t)+\mathbf{C}\mathbf{\dot{q}}(t)+\mathbf{K} \mathbf{q}(t)+\mathbf{b}f(t)=\mathbf{f}_{ext}(t) \tag{1}\]  
where \(t\) is time and \((\dot{\cdot})=d(\ )/dt\) represents the time derivative. For the present building model, the 20 floor displacements are collected in the displacement vector  
\[\mathbf{q}^{T}=[\ q_{1},\ q_{2},\ \ldots,\ q_{20}\ ] \tag{2}\]  
thereby constituting the governing degrees of freedom (dofs) for the present building model, which corresponds to the model used by [PERSON] et al. (1998); [PERSON] et al. (1998). In (1), \(\mathbf{M}\) is the mass matrix, \(\mathbf{C}\) is the damping matrix, and \(\mathbf{K}\) is the stiffness matrix. The construction of the building model is presented in ([PERSON], 1998; [PERSON] et al., 2000) and summarized in Section 4. The external loading is represented by the vector process \(\mathbf{f}_{ext}(t)\), which in Figure 1a will be assumed to be evenly distributed across the building floors. Finally, \(f(t)\) represents the force exerted by the inerter damper onto the structure, with the particular location of the damper determined by the connectivity vector \(\mathbf{b}\). For the location between the ground and first floor in Figure 1, the connectivity vector is given as  
\[\mathbf{b}^{T}=[\ 1,\,0,\,\ldots,\,0\ ] \tag{3}\]  
with entries 2 to 20 all zero.  
In the following damper design and tuning analysis, the behavior of the damper is conveniently represented in the frequency domain, in which time derivatives are included by 90-degree phase shifts. Thus, the time-based equation of motion (1) is conveniently converted into the frequency domain by introducing the steady-state frequency representations  
\[\mathbf{q}(t)=\mathbf{q}\exp(i\omega t)\qquad,\qquad f(t)=f\exp(i\omega t) \qquad,\qquad\mathbf{f}_{ext}(t)=\mathbf{f}_{ext}\exp(i\omega t) \tag{4}\]where the frequency variables \(\mathbf{q}\), \(f\), and \(\mathbf{f}_{ext}\) represent steady-state amplitudes and thus are written without the time argument. By substitution of (4) into (1), the structural equation of motion can be expressed as a frequency relation in the following form  
\[\Big{(}-\omega^{2}\mathbf{M}+i\omega\mathbf{C}+\mathbf{K}\Big{)}\mathbf{q}+ \mathbf{b}f=\mathbf{f}_{ext} \tag{5}\]  
where the damper force \(f\) is defined in the following sections based on the inerter damper considered in the present paper. The frequency solution enables tuning based on both frequency response curves or pole-placement techniques. As shown for the classic tuned mass damper (TMD) in [9] and for general control methods in [43], a balanced pole-placement method will also provide effective response amplitude mitigation in frequency response curves. Thus, the pole-placement technique is applied in Section 3 for inerter damper tuning.