## 5 Discussion
### About the Analysis of the Modal Damping Ratio
#### 5.5.4 Relationship between Modal Damping Ratios and Natural Frequencies  
To conclude this review of the issues related to the dynamic behaviour of the structures that should be studied, another relevant element is the analysis of the possible relationship between the recorded modal damping ratios and the natural frequencies.  
In this sense, it is of interest to check whether the experimental data obtained fit the hypothesis proposed by [PERSON] or not [38; 39]. If damping is supposed to be proportional to the mass and stiffness matrices of the structure, the following expression for the \(i\)-th modal damping ratio of the system can be derived:  
\[\xi_{i}=\frac{1}{2}\left(\frac{\alpha_{0}}{\omega_{i}}+\alpha_{1}\omega_{i} \right)=\frac{\alpha_{0}}{4\pi f_{i}}+\pi\alpha_{1}f_{i}\, \tag{1}\]  
where \(\alpha_{0}\) and \(\alpha_{1}\) are the proportionality parameters.  
Based on the \(N\) experimental data of natural frequencies and associated modal damping ratios, the best possible adjustment would be the one that minimizes the sum of the squared differences between the estimated modal damping factors \(\xi_{i}\) and those measured experimentally. This is equivalent to optimizing the following function:  
\[G(\alpha_{0},\alpha_{1})=\sum_{i}\left(\xi_{i}-\xi_{i}\right)^{2}. \tag{2}\]  
The condition of the best adjustment to minimize the function \(G\) is that the partial derivatives with respect to each of the parameters are null. Thus, combining the two previous expressions, we have:  
\[\min[G(\alpha_{0},\alpha_{1})]\Rightarrow\left\{\begin{array}{l}\frac{ \partial G}{\partial\alpha_{0}}=\sum_{i}\left(\frac{\alpha_{0}}{8\pi^{2}f_{i} ^{2}}+\frac{\alpha_{1}}{2}-\frac{\xi_{i}}{2\pi f_{i}}\right)=0\\ \frac{\partial G}{\partial\alpha_{1}}=\sum_{i}\left(\frac{\alpha_{0}}{2}+2\; \pi^{2}f_{i}^{2}\alpha_{1}-2\pi f_{i}\xi_{i}\right)=0\end{array}\right.. \tag{3}\]  
Solving the system of equations, we arrive at:  
\[\left\{\begin{array}{l}\alpha_{1}=\frac{\frac{\pi^{1}}{N}\sum_{i}(f_{i}\xi_ {i})\sum_{i}\left(\frac{1}{f_{i}^{2}}\right)-\frac{1}{N}\sum_{i}\left(\frac{ \xi_{i}}{f_{i}}\right)}{\frac{1}{N}\sum_{i}\left(f_{i}^{2}\right)\Sigma_{i} \left(\frac{1}{f_{i}^{2}}\right)-N}\\ \alpha_{0}=\frac{4\,\pi}{N}\left(\sum_{i}(f_{i}\xi_{i})-\pi\;\alpha_{1}\sum_{ i}f_{i}^{2}\right)\end{array}\right.\, \tag{4}\]  
These expressions may be directly applied to the experimental data.  
As an example, if this formulation is applied to the data obtained after the field tests of the E-1 structure, coefficients \(\alpha_{0}\) = 0.02272472 and \(\alpha_{1}\) = 0.00020197 result. The recorded modal damping factors (green points), their average values (blue points), and the adjustment obtained by applying the formulation indicated in this section (orange curve) have been represented in Figure 8. It can be seen how the values obtained with the instrumentation reasonably adjust in this case to the Rayleigh hypothesis.  
Since accelerograms in E-2 are extremely short, as mentioned above, the results for the modal damping ratio are not as clear as in E-1, so it was not possible to carry out a similar study.