# Wind-Induced Response Control of High-Rise Buildings Using Inerter-Based Vibration Absorbers
## 2 Equations of Motion and Preliminary Concepts  
According to the sketch in Figure 1, let us consider a high-rise building modeled as a lumped-mass system, equipped with a linear TMDI comprising an attached mass \(m_{t}\) that is connected to the primary structure via linear spring and dashpot elements, \(k_{t}\) and \(c_{t}\), respectively. The TMDI mass is placed in series with one or several linear inerters (of inertance \(b\) in sum) whose two terminals are denoted as 1 and 2, respectively. The first terminal 1 is connected to the attached mass \(m_{t}\), while the second terminal 2 can be attached to another floor along the building height. The building is subject to a wind-induced excitation field represented by a set of forces acting at each floor of the discretized system. Adopting a standard matrix-vector notation, the equations of motion of this \(n\)-story lumped-mass system can be written in the following form  
\[\mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\ddot{\mathbf{u}}(t)+\mathbf{Ku}(t) =\mathbf{p}(t) \tag{1}\]  
where \(\ddot{\mathbf{u}}(t),\dot{\mathbf{u}}(t),\mathbf{u}(t)\) represent the acceleration, velocity, and displacement vectors, respectively, and \(\mathbf{p}(t)\) is the vector of the corresponding aerodynamic forces applied at the center of mass of each floor slab. All these vectors are \((n+1)\)-dimensional, as this is the total number of degrees of freedom (DOFs) of the \(n\)-story building equipped with TMDI. We assume all these vectors are augmented by one last (bottom) row containing the kinematic terms related to the TMDI DOF for \(\ddot{\mathbf{u}}(t),\dot{\mathbf{u}}(t),\mathbf{u}(t)\) and containing a zero entry for the aerodynamic force vector \(\mathbf{p}(t)\). Aerodynamic loading can be obtained from wind tunnel test measurements by synchronous multi-point scanning of pressures on the high-rise building model. In Equation (1), \(\mathbf{M},\mathbf{C},\mathbf{K}\) are the mass, damping and stiffness matrices of the TMDI-equipped structure, respectively. When the TMDI is installed at the \(t^{\text{th}}\) floor and has a so-called \"\(-p\)\" topology (meaning that the second terminal of the inerter is attached to a floor \(t-p\)), these matrices can be expressed as:  
\[ \begin{array}{l}\mathbf{M}=\mathbf{M}_{\text{s}}^{n+1}+(m_{t}+b)\mathbf{1_{n +1}}^{T}\mathbf{1}_{n+1}^{T}+b\mathbf{1_{t-p}}^{T}\mathbf{1}_{t-p}-b\big{(} \mathbf{1_{n+1}}\mathbf{1}_{t-p}^{T}+\mathbf{1_{t-p}}\mathbf{1}_{n+1}^{T} \big{)}\\ \mathbf{C}=\mathbf{C}_{\text{s}}^{n+1}+c_{\text{f}}\big{(}\mathbf{1_{n+1}} \mathbf{1}_{n+1}^{T}+\mathbf{1_{t}}\mathbf{1}_{t}^{T}-\mathbf{1_{n+1}} \mathbf{1}_{t}^{T}-\mathbf{1_{t}}\mathbf{1}_{n+1}^{T}\big{)}\\ \mathbf{K}=\mathbf{K}_{\text{s}}^{n+1}+k\big{(}\mathbf{1_{n+1}}\mathbf{1}_{n+1} ^{T}+\mathbf{1_{t}}\mathbf{1}_{t}^{T}-\mathbf{1_{n+1}}\mathbf{1}_{t}^{T}- \mathbf{1_{t}}\mathbf{1}_{n+1}^{T}\big{)}\end{array}  \tag{2}\]  
where \(\mathbf{M}_{\text{s}}^{n+1},\mathbf{C}_{\text{s}}^{n+1},\mathbf{K}_{\text{s}} ^{n+1}\in\mathbb{R}^{(n+1)\times(n+1)}\) represent the augmented mass, damping and stiffness matrices of the primary structure, respectively, constructed by adding one last (bottom) row with zero entries and one last (rightmost) column of zero entries in the original matrices \(\mathbf{M}_{\text{s}},\mathbf{C}_{\text{s}},\mathbf{K}_{\text{s}}\in\mathbb{R }^{n\times n}\). All the vectors \(\mathbf{1_{j}}\in\mathbb{R}^{(n+1)\times 1}\) are constructed such that only the \(j^{\text{th}}\) entry is equal to one while all the remaining entries are equal to zero (the superscript \(T\) indicates transpose operator). The above equations also apply to a TMD-equipped structure (without inerter), which is retrieved by setting \(b=0\) in the mass matrix. In the same way, a TID scheme as series-Parallel Layout 1 Inerter system proposed in [50] can be obtained by setting \(m_{t}=0\) (without attached mass).