# Wind-Induced Response Control of High-Rise Buildings Using Inerter-Based Vibration Absorbers
## 2 Equations of Motion and Preliminary Concepts
Besides the \((t-p)^{\text{th}}\) entry in the diagonal of the augmented mass matrix, the presence of the inerter modifies the mass matrix by introducingcertain non-diagonal inertial coupling terms between the \(\left(n+1\right)^{\text{th}}\) DOF of the attached mass and the DOF of the \(\left(t-p\right)^{\text{th}}\) floor. Inerter topologies in which the inerter spans more than one story (\(p>1\)) may be realized with pendulum-like implementations like in the Taipei 101 skyscraper (\(p=4\) for 87 th floor to 91 st floor). Such inerter topologies or installation configurations, earlier studied in [36], were found to achieve higher vibration control effectiveness. This was intuitively motivated by the fact that the motion of two non-consecutive floors is seemingly less correlated: therefore, the inerter is likely to undergo higher relative accelerations at its two terminals than if it were installed between two consecutive floors, thus experiencing higher engagement (larger forces for equal inertance value \(b\)). By inspection of Equations (1) and (2) it is worth pointing out the following distinctive aspects in comparison with the earlier study by [PERSON] and [PERSON] [36]: (i) The analysis is here conducted in the time domain since the aerodynamic forces are identified by wind tunnel tests of the benchmark building, whereas [PERSON] and [PERSON] operated on a frequency-domain stochastic approach based on an empirical PSD for the across-wind force field; (ii) The formulation in Equation (2) slightly extends that developed by [PERSON] and [PERSON] since the TMDI should not necessarily be placed at the topmost floor, but it can installed at a generic \(t^{\text{th}}\) floor. Although it is customary to attach the TMD at the top floor because of its widely recognized effectiveness to control the fundamental mode in multistory buildings [58], this installation configuration might be not feasible in some practical projects because of potential structural or architectural constraints. The installation of traditional TMD at different floors, not just at the topmost floor, was recently investigated by [PERSON] and [PERSON] [29]. For TMDI, [PERSON] et al. [59] and [PERSON] and [PERSON] [48] also assessed the influence of installation floor on its performance, although they did not explicitly consider the installation floor as one of the variables to be optimized. Additionally, the inerter might be unable to exert its due performance because the relative peak acceleration does not always occur between the top floor and the \(\left(n-p\right)^{\text{th}}\) floor. This is why in this study we have directly set the installation floor as one of the design variables of the  
Figure 1: Sketch of Tuned Mass Damper-Inerter (TMDI)-equipped high-rise building modeled as a linear lumped-mass system. inerterter-based vibration absorber within parametric optimizations, which represents another novel aspect in comparison with definitions of optimization problem in previous literature studies. The TMDI-equipped structure may have non-proportional damping. Therefore, it is convenient to transform Equation (1) into state space variable form that is more suitable for complex modal analysis  
\[\mathbf{A}\dot{\mathbf{z}}(t)+\mathbf{B}\mathbf{z}(t)=\mathbf{f}(t) \tag{3}\]  
where \(\mathbf{z}(t)=\left[\mathbf{u}(t),\dot{\mathbf{u}}(t)\right]^{T}\) is the state space variable vector and the matrices \(\mathbf{A},\mathbf{B}\) and the vector \(\mathbf{f}(t)\) are expressed as  
\[\mathbf{A}=\left[ \begin{array}{cc}\mathbf{C}&\mathbf{M}\\ \mathbf{M}&0\end{array} \right]\text{, }\mathbf{B}=\left[ \begin{array}{cc} \mathbf{K}&0\\ 0&-\mathbf{M}\end{array} \right]\text{, }\mathbf{f}(t)=\left[ \begin{array}{c} \mathbf{p}(t)\\ 0\end{array} \right]\text{.} \tag{4}\]  
From Equations (3) and (4), after performing complex modal analysis it is possible to determine the transfer function of the system response to assess wind-induced response mitigation induced by the TMD\(\backslash\)TMDI\(\backslash\)TID.