## 2 Methodology Steps
### Theoretical Background  
For an \(N\times N\) multi-degree-of-freedom system, the equation of motion without damping in free vibration because of an initial displacement or velocity is as follows:  
\[\mathbf{m}\ddot{\mathbf{u}}(\mathbf{t})+\mathbf{k}_{o}\mathbf{u}(\mathbf{t})=0 \tag{1}\]  
In Equation (1), \(\mathbf{m}\) and \(\mathbf{k}_{o}\) are, respectively, the \(N\times N\) matrices of the mass and the stiffness of the structure in the health (without damage) state, and \(\mathbf{u}(\mathbf{t})\) and \(\ddot{\mathbf{u}}(\mathbf{t})\) are,respectively, the time \(N\times 1\) varying vectors of displacements and accelerations of the system. Next, it is assumed that the multi-degree-of-freedom system is an existing planar, steel, braced frame with beams rigidly connected to columns, which presents a damage image due to various causes. Therefore, the stiffness matrix at any step \(i\) in the nonlinear area of behavior will change by \(\Delta\mathbf{k}_{i}\). At the same \(i\)-step, the stiffness matrix \(\mathbf{k}_{i}\) of the steel-braced frame is written as  
\[\mathbf{k}_{i}=\mathbf{k}_{o}-\Delta\mathbf{k}_{i} \tag{2}\]  
where \(\Delta\mathbf{k}_{i}\) is the damage stiffness matrix. Correspondingly, the equation of motion (Equation (1)) of the steel-braced frame at the inelastic \(i\)-step is written as  
\[\mathbf{m}\ddot{\mathbf{u}}(\mathbf{t})+(\mathbf{k}_{o}-\Delta\mathbf{k}_{i}) \mathbf{u}(\mathbf{t})=0 \tag{3}\]  
Next, the instantaneous mode shapes \(\boldsymbol{\varphi}_{i,g}\) of the steel-braced frame at various inelastic \(i\)-steps can be found by performing a modal analysis using the instantaneous stiffness matrix \(\mathbf{k}_{i}\), which includes the effects of damage in stiffness. These are vectors of order \(N\times 1\), where \(g=1\), \(2\), \(3\), \(\ldots\), \(N\) in an \(N\)-degree of freedom system. Assuming that the mass matrix does not vary, the eigenvalue problem at the inelastic \(i\)-step can be written as  
\[\Big{[}(\mathbf{k}_{o}-\Delta\mathbf{k}_{i})-\omega_{i,g}^{2}\mathbf{m}\Big{]} \boldsymbol{\varphi}_{i,g}=0 \tag{4}\]  
where \(\omega_{i,g}\) (rad/s) is the instantaneous cyclic frequency (rad/s) of each mode shape at the inelastic \(i\)-step. The solution of the eigenvalue problem is achieved by setting the determinant of the terms inside brackets in the left side of Equation (4) equal to zero and finding the roots \(\omega_{i,g}^{2}\) of the respective algebraic equation:  
\[\text{det}\Big{[}(\mathbf{k}_{o}-\Delta\mathbf{k}_{i})-\omega_{i,g}^{2} \mathbf{m}\Big{]}=0 \tag{5}\]  
The instantaneous mode shape vector \(\boldsymbol{\varphi}_{i,g}\) can be found subsequently from Equation (4) for each one eigenvalue \(\omega_{i,g}^{2}\). Moreover, by knowing an eigenvalue \(\omega_{i,g}^{2}\), Equation (4) is pre-multiplied with the eigenvector \(\boldsymbol{\varphi}_{i,g}^{T}\):  
\[\boldsymbol{\varphi}_{i,g}^{T}\Big{[}(\mathbf{k}_{o}-\Delta\mathbf{k}_{i})- \omega_{i,g}^{2}\mathbf{m}\Big{]}\boldsymbol{\varphi}_{i,g}=0 \tag{6}\]  
Rearranging the terms in Equation (6) leads to  
\[\boldsymbol{\varphi}_{i,g}^{T}\Delta\mathbf{k}_{i}\boldsymbol{\varphi}_{i,g} =\boldsymbol{\varphi}_{i,g}^{T}\mathbf{k}_{o}\boldsymbol{\varphi}_{i,g}- \omega_{i,g}^{2}\boldsymbol{\varphi}_{i,g}^{T}\mathbf{m}\boldsymbol{\varphi}_{ i,g} \tag{7}\]  
By keeping the stiffness and mass terms separate on both sides, Equation (6) or (7) is written as follows in condensed form:  
\[K_{i,g}=\omega_{i,g}^{2}M_{i,g} \tag{8}\]  
where \(K_{i,g}\) and \(M_{i,g}\) are the modal generalized stiffness and modal generalized mass of the \(g\)-mode, respectively, at the inelastic \(i\)-step.