## 5 Discussion
### About the Analysis of the Modal Damping Ratio
#### 5.5.3 About the Calculation Method of the Modal Damping Ratio  
Another relevant issue is knowing which calculation method is used to determine the modal damping ratio and maintaining the same criterion throughout the monitoring period.  
As is well known, there are multiple calculation methods that can be classified according to whether they require controlling the excitation or not and, within the latter, whether they are applicable to specific events or to continuous records.  
Figure 6: Multiple correlation model for natural vibration frequencies.  
The methods of amplification at resonance, of the bandwidth on the amplitude-frequency curve, and of the energy loss per cycle at resonance are among the calculation methods that are required to control the excitation. However, these three methods are complex to apply on real structures because they require the performance of a test whose implementation is not easy on structures in service since they require stopping the traffic and precisely controlling the excitation source, and lastly, but possibly most important, exciting the structure with frequencies close to the resonance frequency, which poses the serious possibility of causing structural damage.  
On the contrary, calculation methods that do not require controlling the excitation, that is, those that study ambient vibrations, avoid the problems indicated above. Within this set, among those that are applicable to records of specific dynamic events, we can mention the logarithmic decrement method, the bandwidth method over the signal spectrum with its two possible formulations (half-power and half-quadratic gain), and the adjustment to the theoretical curve. Likewise, among the methods that are applicable to continuous records, we can mention the random decrement technique (RDT) and the stochastic subspace identification (SSI).  
However, for the same dynamic event, the results of the modal damping ratio obtained by applying the different calculation methods can be very different. As an example, Figure 7 shows an accelerogram corresponding to a specific dynamic event of the E-1 structure (event recorded on 20 August 2010 at 7:55 p.m., selected channel C-5, corresponding to the vertical accelerometer installed in the haunch of the arch, \(f_{\text{exp}\,5}=1.675\) Hz), its study in the frequency domain, the filtered wave, and the values of the modal damping ratio obtained as a result of applying different calculation methods. As can be seen, there is an important variation in the results, which in this case ranges from the 0.18% value obtained through the methods of logarithmic decrement, adjustment to the theoretical curve, and the RDT combined with the logarithmic decrement, to the value of 0.30% obtained through the RDT method combined with curve fitting. Therefore, in SHM systems, it is essential to know which modal damping ratio calculation method has been used and to maintain the same criterion throughout the monitoring period.  
Figure 7: Calculation of the modal damping ratio in a dynamic event of the E-1 structure: (**a**) Registered accelerogram; (**b**) Spectrum; (**c**) Filtered accelerogram; (**d**) Modal damping ratios obtained by applying different calculation methods.  
In line with what is indicated in the preceding paragraphs, the authors are finalizing a specific study, both theoretical and experimental, where the possibility of introducing improvements in calculation methodologies is also assessed [40].