## 1 Introduction
\begin{array}{cccccccc}k_{1}+k_{2}&-k_{2}&0&0&\cdots&\cdots &0\\ -k_{2}&k_{2}+k_{3}&-k_{3}&0&\cdots&\cdots&0\\ 0&-k_{3}&k_{3}+k_{4}&-k_{4}&\cdots&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\cdots&\cdots&\vdots\\ \vdots&\vdots&\vdots&-k_{n-1}&k_{n-1}+k_{n}&-k_{n}&0\\ \vdots&\vdots&\vdots&0&-k_{n}&k_{n}+k_{d}&-k_{d}\\ 0&0&\cdots&0&0&-k_{d}&k_{d}\end{array} \right] \tag{4}\]  
It should be noted that matrices (2) to (4) are idealized as tridiagonal stiffness matrices typical of shear frame structures. Nevertheless, the main purpose of these mathematical expressions is to illustrate the inclusion of the TMD effect on a linear structural system with \(n\) horizontal degrees of freedom, and, therefore, it can be extended to other structural models such as 2-dimensional frames derived from actual building numerical models with a static condensation applied to all other vertical and rotational degrees of freedom. Now, the equation of motion of the system is modified to a space-state representation to determine the dynamic response of the linear system. Hence, the state vector \(\mathbf{z}(t)\) is introduced as:  
\[\mathbf{z}(t)=\left\{ \begin{array}{c}\mathbf{x}(t)\\ \dot{\mathbf{x}}(t)\end{array} \right\} \tag{5}\]  
The above-mentioned vector involves the displacements and velocities of the controlled system. By deploying Equation (1), the state-space representation of the structural system controlled by TMD can be written as:  
\[\left\{ \begin{array}{c}\dot{\mathbf{x}}(t)\\ \ddot{\mathbf{x}}(t)\end{array} \right\}=\left[ \begin{array}{cc}0&\mathbf{I }\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array} \right]\left[ \begin{array}{c}\mathbf{x}(t)\\ \dot{\mathbf{x}}(t)\end{array} \right]+\left[ \begin{array}{c}0\\ \mathbf{M}^{-1}\end{array} \right]\cdot-\mathbf{M}\mathbf{1}\ddot{x}_{\tilde{g }}(t) \tag{6}\]  
where \(\mathbf{0}\) and \(\mathbf{I}\) denote a zero matrix and an identity matrix, respectively. Subsequently, Equation (6) is rewritten as:  
\[\dot{\mathbf{z}}(t)=\mathbf{A}z(t)+\mathbf{B}\cdot-\mathbf{M}\mathbf{1}\ddot {x}_{\tilde{g}}(t) \tag{7}\]  
where \(\mathbf{A}\) represents the transition state matrix, and \(\mathbf{B}\) the location adjustment matrix of the external excitation in the structural system:  
\[\mathbf{A}=\left[ \begin{array}{cc}0&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array} \right] \tag{8}\]  
\[\mathbf{B}=\left[ \begin{array}{c}0\\ \mathbf{M}^{-1}\end{array} \right] \tag{9}\]  
Taking inspiration from the tuning process of TMDs on single degree of freedom systems, and to simplify the optimization process, the \(k_{d}\) and \(c_{d}\) parameters can be rewritten as:  
\[k_{d}=\omega_{s}{}^{2}f^{2}m_{d} \tag{10}\]  
\[c_{d}=2\zeta_{d}f\omega_{s}m_{d} \tag{11}\]with \(\omega_{\rm s}\) as the natural frequency of the structural system and \(f\) and \(\zeta_{d}\) as the dimensionless frequency and damping ratios to be optimized. The reference value \(\omega_{\rm s}\) will be assumed in this investigation as the corresponding frequency to the first vibration mode of the structure. Nevertheless, this is just an assumption to reduce significantly the search domain of the variables \(k_{d}\) and \(c_{d}\) since the optimization problem is focused on the tuning problem for TMDs on multi-degree-of-freedom systems.