## 2 Mechanical Model
### State Space Optimal Tuning of Multiple SA-TLGDs Attached to a MDOF-Main System  
In the case of multiple SA-TLGD attached to a MDOF-system, the tuning process is best performed in two steps. At first, the linearized model is tuned with respect to a selected vibration mode of the main system using the classical [PERSON] formulas, presented in Section 3.1. Fine tuning of the absorber parameters for MDOF-systems is best achieved by also considering adjacent vibration modes in a state space representation, by minimizing the weighted squared area of the frequency response function (FRF). Hence, the coupled linearized equations of motion of the seismic and force excited MDOF-main system with multiple SA-TLGDs attached (Equation (18)), are transformed into the state space. Introducing the state space hypervector \(\overset{\rightharpoonup}{z}=\left[\overset{\rightharpoonup}{w}\overset{ \rightharpoonup}{w}\overset{\rightharpoonup}{w}\overset{\rightharpoonup}{u}\right]\) and its time derivative \(\overset{\rightharpoonup}{z}\), renders the first order matrix equation in state space,  
\[\begin{array}{l}\overset{\rightharpoonup}{z}\ =\ \left(\mathbf{A}+\mathbf{B} \ \mathbf{R}\right)\overset{\rightharpoonup}{z}\ -\ \overset{\rightharpoonup}{\varepsilon}_{g}\bar{w}_{g}\ +\ \overset{ \rightharpoonup}{b},\\ \overset{\rightharpoonup}{\varepsilon}_{g}\ =\ \left[\overset{\rightharpoonup}{0} \overset{\rightharpoonup}{0}\ \overset{\rightharpoonup}{0}\ \mathbf{M}_{\mathbf{S}}^{-1}\left(\begin{array}{c}\mathbf{M}\ \overset{\rightharpoonup}{r}_{S}+\mathbf{L}\ \mathbf{M}_{\mathbf{f}}\overset{ \rightharpoonup}{i}\\ \overset{\rightharpoonup}{i}\end{array}\right)\right],\\ \overset{\rightharpoonup}{b}\ =\ \mathbf{E}_{\mathbf{F}}\overset{\rightharpoonup}{F},\ \mathbf{E}_{ \mathbf{F}}\ =\ \left[\mathbf{0}\ \ \ \mathbf{0}\ \ \ \mathbf{M}_{\mathbf{S}}^{-1}\left(\begin{array}{c}\mathbf{I}\\ \mathbf{0}\end{array}\right)\right]\!.\end{array} \tag{25}\]  
The forcing hypervector contains the horizontal excitation forces \(\overset{\rightharpoonup}{F}(t)\). Since only the matrix elements \(\mathbf{A}\) and \(\mathbf{B}\) from Equation (25) contain the known main system parameters, the state space matrix \(\mathbf{A}+\mathbf{B}\ \mathbf{R}\) should be kept separated,  
\[\begin{array}{l}\mathbf{A}\ =\ \left[\begin{array}{ccc}\mathbf{0}\To determine the optimal design parameters of the SA-TLGD it is common practice to minimize an appropriate performance index, e.g., defined by the infinite integral of the weighted sum of quadratic state variables of the MDOF-main system \(\overset{\rightarrow}{z}_{S}\), in the frequency domain, see, e.g., [PERSON] and [PERSON] [43], for the seismic excited coupled system  
\[J(\
u)\ =\ \int\limits_{-\infty}^{\infty}\overset{\rightarrow}{z}_{S} (\
u)\ \mathbf{S}\overset{\rightarrow}{z}_{S}(\
u)\ dv = 2\pi\overset{\rightarrow}{e}_{g}^{T}\mathbf{P}\overset{ \rightarrow}{e}_{g}\quad\rightarrow\ min, \tag{30}\]  
and for the force excited coupled system,  
\[J(\
u)\ =\ \ \int\limits_{-\infty}^{\infty}\overset{\rightarrow}{z}_{S} ^{T}(\
u)\ \mathbf{S}\overset{\rightarrow}{z}_{S}(\
u)\ dv = 2\pi\overset{\rightarrow}{b}_{0}^{T}\mathbf{P}\overset{ \rightarrow}{b}_{g}\quad\rightarrow\ min. \tag{31}\]  
where the matrix \(\mathbf{P}\) is the solution of the algebraic Lypanuov matrix equation,  
\[(\mathbf{A}+\mathbf{B}\ \mathbf{R})^{T}\mathbf{P}\ +\ \mathbf{P}\ (\mathbf{A}+ \mathbf{B}\ \mathbf{R})\ =\ -\mathbf{S}. \tag{32}\]  
The matrix \(\mathbf{S}\) is a symmetric, positive semi-definite weighing matrix, which offers the opportunity to emphasize the importance of selected components of the state space vector. The matrix solution for \(\mathbf{P}\) is numerically evaluated by means of the software MATLAB. The minimum search is best performed by the MATLAB optimization toolbox, substituting [PERSON]'s modal design parameters for the SA-TLGDs as start values.