## 2 Methods
\[\mathbf{q}(t)=\sum_{i=1}^{N}\mathbf{a}_{i}\psi_{i}(t)=\mathbf{W}\psi(t), \tag{2}\]3  
A system of motion equations for such a case, with the number of degrees of freedom of \(N_{k}+1\), is shown by the formula below:  
\[ \begin{array}{c}\overline{m}_{ji}\ddot{\overline{q}}_{ji}(t)+\overline{c}_{ji }\ddot{\overline{q}}_{ji}(t)+\overline{k}_{ji}\overline{q}_{ji}(t)=\overline{ p}_{ji}(t)+T_{ji}\\ m_{k}^{t}\ddot{q}_{k}^{t}+c_{k}^{t}\left(\dot{q}_{k}^{t}-\dot{\tilde{q}}_{k}^{ t}\right)+k_{k}^{t}\left(q_{k}^{t}-\widetilde{q}_{k}^{t}\right)=0\end{array}  \tag{12}\]  
The force from each damper, which needs to be attached to the degree of freedom associated with the \(k\)-th TMD, taking into account absolute values, can be expressed with the following formulas:  
\[T_{k}=-m_{k}^{t}\ddot{q}_{k}^{t}, \tag{13}\]  
\[T_{k}=c_{k}^{t}\left(\dot{q}_{k}^{t}-\dot{\tilde{q}}_{k}^{t}\right)+k_{k}^{t} \left(q_{k}^{t}-\tilde{q}_{k}^{t}\right)=c_{k}^{t}\left(\dot{q}_{k}^{t}-\frac {\widetilde{a}_{ki}}{a_{ji}}\ddot{\overline{q}}_{ji}\right)+k_{k}^{t}\left(q_ {k}^{t}-\frac{\widetilde{a}_{ki}}{a_{ji}}\ddot{\overline{q}}_{ji}\right). \tag{14}\]  
Changing to generalized force, for absolute values we get:  
\[T_{ji}=\frac{1}{a_{ji}}\sum\limits_{k=1}^{N_{k}}\widetilde{a}_{ki}T_{k}=-\sum \limits_{k=1}^{N_{k}}\frac{\widetilde{a}_{ki}}{a_{ji}}m_{k}^{t}\dot{q}_{k}^{ t}, \tag{15}\]  
\[T_{ji}=\frac{1}{a_{ji}}\sum\limits_{k=1}^{N_{k}}\widetilde{a}_{ki}T_{k}=\sum \limits_{k=1}^{N_{k}}\frac{\widetilde{a}_{ki}}{a_{ji}}c_{k}^{t}\left(\dot{q}_ {k}^{t}-\frac{\widetilde{a}_{ki}}{a_{ji}}\ddot{\overline{q}}_{ji}\right)+ \sum\limits_{k=1}^{N_{k}}\frac{\widetilde{a}_{ki}}{a_{ji}}k_{k}^{t}\left(q_{k} ^{t}-\frac{\widetilde{a}_{ki}}{a_{ji}}\overline{q}_{ji}\right). \tag{16}\]  
By substituting the additional damping force values to motion equations, and by using the relationships for absolute values in Equations (15) and (16), we get:  
\[ \begin{array}{c}\overline{m}_{ji}\ddot{\overline{q}}_{ji}(t)+\overline{c}_{ji }\ddot{\overline{q}}_{ji}(t)+\overline{k}_{ji}\ddot{\overline{q}}_{ji}(t)+ \sum\limits_{k=1}^{N_{k}}\frac{\widetilde{a}_{ki}}{a_{ji}^{t}}c_{k}^{t}\ddot{ \overline{q}}_{ji}-\sum\limits_{k=1}^{N_{k}}\frac{\widetilde{a}_{ki}}{\dot{ \overline{q}}_{ji}}c_{k}^{t}\dot{q}_{k}^{t}+\sum\limits_{k=1}^{N_{k}}\frac{ \widetilde{a}_{ki}^{t}}{\dot{\overline{q}}_{ji}}^{2}k_{k}^{t}\overline{q}_{ji} -\sum\limits_{k=1}^{N_{k}}\frac{\widetilde{a}_{ki}^{t}}{\dot{\overline{q}}_{ ji}}k_{k}^{t}\overline{q}_{ji}-\sum\limits_{k=1}^{N_{k}}\frac{\widetilde{a}_{ki}^{t}}{ \dot{\overline{q}}_{ji}}k_{k}^{t}\overline{q}_{ji}=\overline{p}_{ji}\\ m_{k}^{t}\ddot{q}_{k}^{t}+c_{k}^{t}\left(\dot{q}_{k}^{t}-\frac{\widetilde{a}_{ki }}{\dot{\overline{q}}_{ji}}\ddot{\overline{q}}_{ij}\right)+k_{k}^{t}\left(q_{k }^{t}-\frac{\widetilde{a}_{ki}}{\dot{\overline{q}}_{ji}}\overline{q}_{ji} \right)=0\end{array}  \tag{17}\]  
Motion equations in matrix form, for the most general case, with absolute values, are shown by the relationship below:  
\[\overline{\mathbf{M}}\ddot{\overline{\mathbf{q}}}+\overline{\mathbf{C}}\dot{ \overline{\mathbf{q}}}+\overline{\mathbf{K}}\overline{\mathbf{q}}=\overline{ \mathbf{p}}, \tag{18}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are positive-defined mass, damping, and stiffness matrices, respectively (it is impossible to assume negative values of stiffness and damping of individual TMDs). From the generalized displacement and excitation forces' vector let us separate vector blocks associated with the equivalent displacement modal ordinate and actual TMD displacements:  
\[\ddot{\overline{\mathbf{q}}}=\left[ \begin{array}{c}\ddot{\overline{\mathbf{q }}}_{1}\\ \dot{\overline{\mathbf{q}}}_{2}\end{array} \right]\text{, }\dot{\overline{\mathbf{q}}}=\left[ \begin{array}{c}\ddot{\overline{ \mathbf{q}}}_{1}\\ \dot{\overline{\mathbf{q}}}_{2}\end{array} \right]\text{, }\overline{\mathbf{q}}=\left[ \begin{array}{c}\overline{\mathbf{q}}_{1}\\ \overline{\mathbf{q}}_{2}\end{array} \right]\text{, }\overline{\mathbf{p}}=\left[