## 2 Materials and Methods  
### Proportional and Non-Proportional Damping  
The dynamic equilibrium equation of a linear lumped system with \(N\) degrees of freedom under a single component of base earthquake excitation is [17]:  
\[\mathbf{m}\ddot{\mathbf{u}}(t)+\mathbf{c}\dot{\mathbf{u}}(t)+\mathbf{k}\mathbf{ u}(t)=-\mathbf{m}u_{g}(t), \tag{1}\]  
where \(\mathbf{u}\) is the displacement vector, \(\mathbf{m}\) is the mass matrix, \(\mathbf{c}\) is the damping matrix, \(\mathbf{k}\) is the stiffness matrix, \(\mathbf{\iota}\) is the influence vector representing displacements in the lumped story masses due to a unit base displacement, and \(a_{g}(t)\) is the ground acceleration. Equation (1) can be transformed using the undamped system's modal matrix \(\mathbf{\Phi}\) (where each column \(\mathbf{\varphi}_{j}=[\mathbf{\varphi}_{1j},\dots\mathbf{\varphi}_{Nj}]^{T}\), corresponds to the \(j^{\text{th}}\) vibration mode); applying the change of coordinates \(\mathbf{u}(t)=\mathbf{\Phi}\mathbf{y}(t)\) (where \(\mathbf{y}(t)\) is a \(N\times 1\) vector of time-dependent modal coordinates), pre-multiplying by \(\mathbf{\Phi}^{\text{T}}\) and using the orthogonality properties of natural modes, the expression reads:  
\[\mathbf{M}\ddot{\mathbf{y}}(t)+\mathbf{C}\dot{\mathbf{y}}(t)+\mathbf{K}y(t)=- \mathbf{\Phi}^{\text{T}}\mathbf{m}u_{g}(t), \tag{2}\]  
where the diagonal matrices \(\mathbf{M}=\mathbf{\Phi}^{\text{T}}\mathbf{m}\mathbf{\Phi}\) and \(\mathbf{K}=\mathbf{\Phi}^{\text{T}}\mathbf{k}\mathbf{\Phi}\) are the modal mass and modal stiffness matrix, respectively, and \(\mathbf{C}=\mathbf{\Phi}^{\text{T}}\mathbf{c}\mathbf{\Phi}\) is the modal damping matrix. A special case of this transformation occurs when the modal matrix is previously normalized to the system's mass matrix, i.e., \(\mathbf{\hat{\Phi}}^{\text{T}}\mathbf{m}\mathbf{\hat{\Phi}}=\mathbf{I}\) (where \(\mathbf{I}\) is the \(N\times N\) identity matrix and \(\mathbf{\hat{\phi}}_{j}=\mathbf{\phi}_{j}M_{j}^{-0.5}\)), whereupon:  
\[\ddot{y}(t)+\mathbf{\hat{C}}\dot{y}(t)+\mathbf{\Omega}^{2}y(t)=-\mathbf{\hat{ \Phi}}^{\text{T}}\mathbf{m}u_{g}(t); \tag{3}\]In this expression \(\mathbf{\Omega}^{2}\) is the diagonal spectral matrix, with \(j^{\text{th}}\) term \(\omega_{j}^{2}\), where \(\omega_{j}\) is the \(j^{\text{th}}\) mode undamped circular frequency, and \(\mathbf{\hat{C}}\) is referred to as normalized modal damping matrix. If \(\mathbf{\hat{C}}\) is a diagonal matrix, Equation (3) results in \(N\) uncoupled modal equations:  
\[\ddot{y}_{j}(t)+\hat{C}_{j}\dot{y}_{j}(t)+\omega_{j}^{2}y_{j}(t)=-\hat{\Gamma} _{j}a_{g}, \tag{4}\]  
where \(\hat{C}_{j}\) is the \(j^{\text{th}}\) diagonal term of \(\mathbf{\hat{C}}\), and \(\Gamma_{j}\big{(}=\dot{\Phi}_{j}^{\text{T}}\mathbf{m}\big{)}\) is the mass-normalized modal participation factor corresponding to mode \(j\). Each mode is thus represented by an independent SDOF system with modal damping ratio \(\zeta_{j}=\hat{C}_{j}/\big{(}2\omega_{j}\big{)}=C_{j}/\big{(}2M\omega_{j}\big{)}\). As shown by [PERSON] and [PERSON] [6], \(\mathbf{\hat{C}}\) will be diagonal if, and only if:  
\[\mathbf{c}\mathbf{m}^{-1}\mathbf{k}=\mathbf{k}\mathbf{m}^{-1}\mathbf{c}. \tag{5}\]  
This is known as the commutability condition and can be met by a Caughey damping matrix \(\mathbf{c}\) which is a combination of \(\mathbf{m}\) and \(\mathbf{k}\)[18]. Mass-proportional damping, stiffness-proportional damping and Rayleigh damping are all special cases of Caughey damping. If Equation (5) is fulfilled, \(\mathbf{c}\) is referred to as proportional damping matrix, \(\mathbf{\hat{C}}\) is diagonal, and each modal equation given by (4) can be solved independently using time-history modal analysis (THMA) to find the complete modal response for each mode. Addition of modal responses at each time step produces the exact total system response. This is the classical modal method.