## 2 Materials and Methods
### Numerical Study
#### 2.3.1 Systems Analyzed
Finally, the approximate modal damping and damping matrices are obtained preserving only the diagonal terms in \(\mathbf{C}\):  
\[\mathbf{C}_{\text{approx}}=diag(C_{n}), \tag{13}\]  
\[\mathbf{c}_{\text{approx}}=\Big{(}\mathbf{\Phi}^{T}\Big{)}^{-1}\mathbf{C}_{ approx}\mathbf{\Phi}^{-1}. \tag{14}\]  
It is worth noting that modal damping ratios \(\xi_{j}\) (=\(C_{j}/2\omega_{j}M_{j}\)) obtained from \(\mathbf{C}\) and \(\mathbf{C}_{approx}\) are identical. The added damping matrix \(\mathbf{c}_{a}\) is totally defined from \(N\), \(c_{d,1}\), \(\alpha_{d}\), \(\beta_{d}\). However, instead of fixing \(c_{d,1}\), this value is determined to target a certain value of added damping ratio in the first mode \(\xi_{a,1}\) (an initial value \(c_{d,1}\)* is used to assemble the initial modal damping matrix \(\mathbf{C}^{*}\); \(\xi_{a,1}^{*}\) is then calculated from \(C_{1}^{*}\), whereupon \(c_{d,1}=c_{d,1}*\)*\(\xi_{a,1}/\xi_{a,1}^{*}\), and the actual added damping matrix \(\mathbf{c}_{a}\) can be obtained). \(\mathbf{c}_{a}\) is thus a band matrix of bandwidth \(1\). This is also the case for \(\mathbf{k}\), if \(\lambda=\infty\); for other values of \(\lambda\), \(k\) is populated, with increasing off-diagonal terms for lower values of \(\lambda\). The formulation used for the damping system assumes pure viscous behavior and thus the influence of auxiliary element flexibility [39] is not included in the study. As indicated, six different values of \(\lambda\) [0.10; 0.50; 1; 2; 10; \(\infty\)] are explored. \(\xi_{h}\) is generally taken as 3%, although some 0% and 5% cases are also included to determine the influence of this parameter. With \(\lambda\) and \(\xi_{h}\) fixed, the remaining parameters defining a damped system are: \(N\), \(T_{1}\), \(\xi_{a,1}\), \(\alpha_{k}\), \(\beta_{k}\), \(\alpha_{d}\), \(\beta_{d}\). The range considered for these 7 parameters in the study is: \(N\) = [3, 5, 7, 9]; \(\alpha_{k}\) = [0.20; 0.40; 0.60; 0.80]; \(\beta_{k}\) = [0.5; 1; 2; 10]; \(\alpha_{d}\) = [0; 1/3; 2/3; 1]; \(\beta_{d}\) = [0.1; 0.5; 1; 2]; \(T_{1}\) = [0.5; 1; 2; 4]; \(\xi_{a,1}\) = [10%; 20%; 30%; 40%]. The values chosen for \(N\) and \(T_{1}\) cover the most usual cases in building structures equipped with linear viscous dampers, where a certain lateral flexibility is needed to render the damping devices useful. The uppermost 40% value for \(\xi_{a,1}\) is slightly above the current maximum value (\(\xi_{h,1}+\xi_{a,1}=35\%\)) allowed in US practice for application of RSA [8]. The values chosen for \(\alpha_{k}\) correspond to realistic possibilities in a design situation; the values chosen for \(\alpha_{d}\), \(\beta_{d}\), and \(\beta_{k}\) range between extreme (\(\alpha_{d}=[0,1/3]\), \(\beta_{d}=0.1\), \(\beta_{k}=10\)) and usual (\(\alpha_{d}=[2/3,1]\), \(\beta_{d}=[0.5,1,2]\), \(\beta_{k}=[0.5,1,2]\)). The extreme value (\(\alpha_{d}=0\), \(\beta_{d}=0.1\)) represents approximately a structure with additional dampers only in the first story. For \(\alpha_{d}=1\) any value of \(\beta_{d}\) results in the same damping system, thus only \(\beta_{d}=2\) need to be considered, and \(3\cdot 4^{5}\) repeated cases (\(\alpha_{d}=1\) with \(\beta_{d}=0.1\), \(0.5\) and \(1\)) are eliminated from the set. The total number of different systems examined for every combination of \(\lambda\) and \(\xi_{h}\) is therefore 13,312 (\(4^{7}-3\cdot 4^{5}\)). Combinations of \(\alpha_{k}\), \(\beta_{k}\), \(\alpha_{d}\) and \(\beta_{d}\) considered in the study are illustrated in Figure 2.  
\[\frac{k_{c,eq,n}}{k_{c,eq,1}}=1+(\alpha_{k}-1)\bigg{(}\frac{n-1}{N-1}\bigg{)}^ {\beta_{k}}, \tag{8}\]0 Vertical distributions considered in the study; **(a)** stiffness; **(b)** added damping.