## 2 Materials and Methods
### Numerical Study
#### 2.3.1 Systems Analyzed  
The planar elastic MDOF structures included in the study can be described by the generic frame shown in Figure 1a, defined by a vertical distribution of column stiffness \(k_{c,eq,n}\), beam stiffness \(k_{b,eq,n}\), and story mass \(m_{n}=0.25\) kN-s\({}^{2}\)/mm equal at all stories. For any element \(x\) (beam or column), \(k_{eq,x}=EI_{eq,n}\)/\(L_{x}\), where \(E\) is the Young modulus (210 GPa for all elements), \(l_{eq,x}\) is the element's second moment of inertia associated to in-plane displacement, and \(L_{x}\) is the element centerline length. Subscript 'eq' is used because the one-bay frame in Figure 1a can be seen as a surrogate for a multi-bay frame, as described in Appendix B. The axial and shear flexibility of the members is neglected, as is the out-of-plane behavior, and it is assumed that \(h_{n}=h\), \(L=2h\), and the ratio between beam and column stiffness \(\lambda\) is constant for the whole frame:  
\[\lambda_{n}=EI_{b,eq,n}/EI_{c,eq,n}=2k_{b,eq,n}/k_{c,eq,n}=\lambda. \tag{7}\]  
As discussed in Appendix B, values of \(\lambda=0.5\) to 2 are usual in steel seismic resisting frames. In this study, six different values of \(\lambda\) (0.10; 0.50; 1; 2; 10; \(\infty\)) are considered, with the special case \(\lambda_{n}=\infty\) corresponding to so-called shear-type systems, and \(\lambda_{n}<\infty\) corresponding to frame-type systems. The vertical distribution of column stiffness is taken as:  
\[\frac{k_{c,eq,n}}{k_{c,eq,1}}=1+(\alpha_{k}-1)\bigg{(}\frac{n-1}{N-1}\bigg{)}^ {\beta_{k}}, \tag{8}\]  
Figure 1: Parametric definition of MDOF systems: (**a**) frame-type system included in the study; (**b**) linear added damping system; (**c**) damper coefficient distribution; (**d**) column stiffness distribution. where the parameter \(\alpha_{k}\) represents the proportion between roof and first story stiffness, \(\alpha_{k}=k_{c,eq,N}/k_{c,eq,1}\), and \(\beta_{k}\) defines the height-wise variation law of stiffness, see Figure 1d. A similar concept is used to define the added damping coefficients, see Figure 1b,c:  
\[\frac{c_{d,n}}{c_{d,1}}=1+(\alpha_{d}-1)\bigg{(}\frac{n-1}{N-1}\bigg{)}^{\beta_ {d}}. \tag{9}\]  
The frame stiffness matrix is totally defined by \(N\), \(k_{c,eq,1}\), \(\alpha_{k}\), \(\beta_{k}\) and \(\lambda\). However, instead of fixing \(k_{c,eq,1}\), this value is determined to target a certain fundamental period \(T_{1}\) (an initial value \(k_{c,eq,1}\)* is used to assemble the initial stiffness matrix \(\mathbf{k}\)* and mass matrix \(\mathbf{m}\); using these values the period \(T_{1}\)* is calculated, whereupon \(k_{c,eq,1}=k_{c,eq,1}\)*\(T_{1}/T_{1}\)*, and the actual stiffness matrix \(\mathbf{k}\) can be obtained). The undamped vibrational properties of the system, \(\mathbf{\Phi}\) and \(\omega_{n}\), are calculated from \(\mathbf{k}\) and \(\mathbf{m}\). The matrix \(\mathbf{k}\) is condensed to include only translational terms [17] (pp. 367 ff.). The damped vibrational properties are determined as follows: first, an inherent modal damping ratio \(\xi_{h}\) is assumed for all modes, the diagonal inherent modal damping matrix is assembled as \(C_{h,j}=2\xi_{h}\omega_{j}M_{j}\), and the corresponding inherent damping \(\mathbf{c}_{h}\) is calculated as:  
\[\mathbf{c}_{h}=\Big{(}\mathbf{\Phi}^{T}\Big{)}^{-1}\mathbf{C}_{h}\mathbf{\Phi }^{-1}. \tag{10}\]  
Second, the added damping matrix \(\mathbf{c}_{a}\) is obtained from first principles:  
\[\mathbf{c}_{a}=\mathbf{T}_{U}{}^{-1}\mathbf{c}_{d}\mathbf{T}_{L}{}^{-1}, \tag{11}\]  
where \(\mathbf{T}_{U}\) and \(\mathbf{T}_{L}\) are the \(N\)-dimensional unity upper and lower triangular matrices, respectively, and \(\mathbf{c}_{d}=\text{diag}(c_{d,n})\) where \(\text{diag}(x_{n})\) returns a diagonal matrix with \(n\)-th term \(x_{n}\), whereupon the total damping matrix is found as:  
\[\mathbf{c}=\mathbf{c}_{h}+\mathbf{c}_{a}, \tag{12}\]  
and the total modal damping matrix is calculated as \(\mathbf{C}=\mathbf{\Phi}^{T}\mathbf{c}\mathbf{\Phi}\).