## 7 Conclusions
\begin{array}{cc}m_{e}&\underline{0}\\ \underline{M}\,e&\underline{\underline{M}}\,\left(\!\!\begin{array}{c}1\times N \\ \underline{M}\,e\end{array}\!\!\begin{array}{c}1\times N\\ \underline{M}\,\overline{N}\end{array}\!\!\right)\end{array} \right]\!\left[ \begin{array}{cc}\begin{array}{c}\bar{d}_{b}\\ \underline{\underline{\Delta}}\bar{d}\end{array}\!\!\end{array} \!\!\right]+ \left[ \begin{array}{cc}0&-\underline{e}^{T}\underline{\underline{K}}\\ \underline{0}&\underline{\underline{K}}\\ (N\times 1)\end{array} \right]\!\left[ \begin{array}{c}d_{b}\\ \underline{\underline{\Delta}}\bar{d}\end{array} \!\!\right]=\left[ \begin{array} []{c}0\\ \underline{0}\\ (N\times 1)\end{array} \right]. \tag{10}\]  
Since \(\bar{d}_{b}\) is equal to the seismic accelerogram (a(t)), the second part of Equation (10) can be rewritten as follows:  
\[\underline{\underline{M}}\,\underline{\underline{\Delta}}\bar{d}+\underline{K }\,\underline{\Delta}\bar{d}=-\underline{M}\,\underline{\underline{\varepsilon }}\,\underline{\underline{\varepsilon}}\,(t) \tag{11}\]  
which allows for the evaluation of the seismic response of the structure (i.e., of the time histories of the displacements (\(\underline{\Delta}\underline{d}\)(/speeds \(\underline{\Delta}\underline{d}\)/accelerations \(\underline{\Delta}\underline{d}\))) of the \(\mathrm{I}_{\mathrm{S}}\) nodes with respect to the base of S). In the presence of viscous dissipation, all the considerations made so far may be repeated, which thus leads to the following expression of the equation that governs the seismic response of the structure:  
\[\underline{\underline{M}}\,\underline{\underline{\Delta}}\bar{d}+\underline{ \underline{\varepsilon}}\,\underline{\Delta}\underline{d}+\underline{K}\, \underline{\Delta}\bar{d}=-\underline{M}\,\underline{\underline{\varepsilon }}\,\underline{\underline{\varepsilon}}\,(t) \tag{12}\]  
where \(\underline{\underline{\varepsilon}}\) is the viscous damping matrix of the system and, for the cases of  
where \(h_{i}(t)\) is the impulse response function of the \(i\)-th generalized coordinate:  
\[h_{i}(t)=\frac{e^{-\zeta_{i}\omega_{Ni}t}\sin(\omega_{Di}t)}{\omega_{Di}}\]  
where:  
\(\omega_{Ni}\) is the natural pulsation of the \(i\)-th mode;  
\(\zeta_{i}=\frac{\sigma_{i}}{2m_{G}\omega_{Ni}}\) is the modal damping of the \(i\)-th mode;  
\(\omega_{Di}\) is the damped pulsation of the \(i\)-th mode, which is equal to \(\omega_{Ni}\sqrt{1-\zeta_{i}^{2}}\). Once the time histories of the generalized coordinates are evaluated, together with their first and second derivatives, the seismic response of the structure can be obtained by recalling that:  
\[\underline{\Delta d}=\underline{\Phi}_{V}\underline{q}(t),\ \underline{\Delta d}= \underline{\Phi}_{V}\underline{\dot{q}}(t),\ \underline{\Delta d}=\underline{\Phi}_{V}\underline{\dot{q}}(t). \tag{10}\]  
The evaluation of the time histories of _all_ the generalized coordinates may, however, represent a demanding task from a computational standpoint, especially when \(N\) is significantly large. This is why, in general practice, only a subset of modes (and generalized coordinates) is selected to solve the dynamic equation of the structure with adequate approximation. Clearly, the greater the number of selected modes, the better the level of approximation will be. On the other hand, not all modes make the same contribution to the physical response of the structure, and, after a certain number of modes, the computational effort is not rewarded by a related increase in the results' precision. The residual inertia criterion represents a powerful tool to individuate the right number of modes to be used for the evaluation of the seismic response with a good level of approximation.