## 4 Numerical Studies on the Effectiveness of SA-TLGDs
### MDOF-Shear Frame Structure with Multiple SA-TLGDs Attached  
The effectiveness of multiple SA-TLGDs in vibration reduction is demonstrated for a plane three-DOF-shear frame structure under horizontal force excitation, e.g., caused by wind gusts. Based on a benchmark definition paper from [PERSON] et al. [47], a scale model of the original structure was built at the Multidisciplinary Center for Earthquake Engineering Research (MCEER) at Buffalo. In the following numerical study, the considered force excited MDOF-shear frame model with a total mass of 2943 kg and a total height of 2.55 m is equipped with two SA-TLGDs in parallel connection on the 3 rd floor (see Figure 8). The optimization procedure of multiple vibration absorbers that are supposed to be tuned to the most critical resonance frequency of the structure, requires fine-tuning in the state space (see Section 3.2). Modal (SDOF) tuning as discussed in Section 3.1 is performed in a first step. The mass- and stiffness matrix of the scale MDOF-shear frame model are provided in [48],  
\[\mathbf{M}\;=\;\left[ \begin{array}{ccc}981&0&0\\ 0&981&0\\ 0&0&981\end{array} \right]\left[\text{kg}\right]\text{. }\;\mathbf{K}\;=\; \left[ \begin{array}{ccc}650.3&-183.4&33.2\\ -183.4&574.7&-148.9\\ 33.2&-148.9&387.2\end{array} \right]\left[\text{N}/\text{m}\right]\text{,} \tag{37}\]  
and the orthonormalized eigenvectors and well-separated undamped natural frequencies are yielded after solving the eigen value problem [39],  
\[\begin{array}{c}\rightharpoonup\\ \rightharpoon  
With \(\mu\) = 4%, the fluid mass of each attached SA-TLGD results in \(m_{f1}\) = \(m_{f2}\) = 30 kg, i.e., the total fluid mass is \(m_{f,total}\) = 60 kg. In a first step, the optimal absorber eigenfrequencies \(f_{A1,opt},f_{A2,opt}\), and optimal linearized viscous damping ratios \(\zeta_{A1,opt}\), \(\zeta_{A2,opt}\) of the two attached SA-TLGDs are determined by the application of [PERSON]'s formulas. Inserting \(\mu\) = 4% into Equation (22) yields identical optimal design parameters for both SA-TLGDs: \(f_{A1,opt}\) = \(f_{A2,opt}\) = 2.29 Hz, \(\zeta_{A1,opt}\) = \(\zeta_{A2,opt}\) = 12.01%. Improvements in their performance are achieved by minimizing the frequency domain-based quadratic performance index \(J(\
u)\) in the state space representation (see Section 3.2). The state vector of the MDOF-shear frame structure, \(\overset{\rightarrow}{z}_{S}=[w_{1},\ w_{2},\ w_{3},\ \dot{w}_{1},\ \dot{w}_{2},\ \dot{w}_{3}]^{T}\), to be substituted in Equation (31), does not contain the SA-TLGD quantities explicitly. However, the vibration damping effects of the attached SA-TLGDs are hidden in the system's dynamics, Equation (29), and thus in the structural response state vector \(\overset{\rightarrow}{z}_{S}\). The relevant system's matrices \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{R}\) are defined in Equations (26) and (27) and they contain both, the MDOF-main system's parameters and the quantities of the two attached SA-TLGDs. Having chosen the weighing matrix, \(S=diag[10,\ 10,\ 10,\ 1,\ 1,\ 1]\) in Equation (32), the numerical minimization of the performance index \(J(\
u)\) is started with [PERSON]'s modal tuning parameters as initial values. Calling the command _frinusearch_ within the MATLAB optimization toolbox renders the final improved optimal tuning parameters for the two attached SA-TLGDs, significantly changed to: \(f_{A1,opt}\) = 2.18 Hz, \(f_{A2,opt}\) = 2.49 Hz, and \(\zeta_{A1,opt}\) = 6.08%, \(\zeta_{A2,opt}\) = 6.72%. It is noticed that the eigenfrequencies \(f_{A1,opt}\) and \(f_{A2,opt}\) are significantly smaller and larger, respectively, than the considered fundamental natural frequency of the MDOF-shear frame structure \(f_{S1}\) = 2.38 Hz. This gained properties regarding the eigenfrequencies of the two SA-TLGDs to increase the robustness of the  
Figure 8: Scaled three-DOF-shear frame structure under horizontal force excitation \(F_{3}(t)\) with two SA-TLGDs in parallel connection attached to the 3 rd floor.