# Dynamic Identification of Tensile Force in Tie-Rods by Interferometric Radar Measurements
## 5 Estimation of Tensile Force Starting from Vibration Data
### Experimental Identification of Natural Frequencies  
For the modal identification process, interferometric radar data were processed by using the custom IBIS software; in particular, the peak picking method [48] was applied based on the power density spectrum; therefore, only natural frequencies were estimated. In Table 4, the first three natural frequencies \(f_{r}\) estimated in the presence of the corner  
Figure 6: Modified structural model with a lumped mass. reflector starting from interferometric radar data are reported for an applied tensile force of 50 KN. For the sake of the comparison, also the first three natural frequencies \(f_{a}\) estimated in the presence of the corner reflector starting from accelerometric data are reported in the same table. In the latter case, the frequencies were determined by performing frequency domain decomposition (FDD) [49; 50] through a MATLAB routine [51]. Notice that it was impossible to apply the more accurate FDD approach to interferometric radar data since, for FDD, information on multiple measurement points is needed, whereas the only usable information acquired by the radar interferometer is that of the measurement point where the corner reflector was installed. Tables 5 and 6 reported both the first three natural frequencies \(f_{r}\) and \(f_{a}\) determined starting from interferometric radar and accelerometric data, for the tie-rod with installed the corner reflector, and for a known tensile force of 100 KN (Table 5) and 150 kN (Table 6). Tables 4-6 also reported the relative differences between the interferometric radar and accelerometric determined frequency. It is easily seen that the mean value of the frequency discrepancy between interferometric radar and accelerometric data is 0.246%, and its maximum value is 1.187%. This validates the interferometric radar technique and shows that if referring to tensile force estimation approaches, which require only the knowledge of natural frequencies, interferometric radar measurements can be considered equivalent to accelerometric measurements. Of course, this equivalence holds when the corner reflector is installed. Another comment concerns the impossibility of identifying the frequency of the third mode from interferometric radar measurements: this is because this frequency is more than double the employed sampling frequency and thus is inaccessible to the device. Anyway, in practical applications, the tie-rods installed in historic construction are usually characterized from lower frequencies compared to that in Table 6, and therefore, the above-described issue of the radar interferometer does not constitute an actual limit. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode Number**} & \multicolumn{3}{c}{**150 kN**} \\ \cline{2-4}  & \(f_{r}\) **(Hz)** & \(f_{a}\) **(Hz)** & \(f_{r}\)-\(f_{a}\)/\(f_{r}\) **(\%)** \\ \hline
1 & 44.345 & 44.238 & 0.242 \\
2 & 121.726 & 121.875 & 0.122 \\
3 & - & 151.753 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Natural frequencies experimentally determined starting from interferometric radar data (\(f_{r}\)) and accelerometric data (\(f_{a}\)). Applied tensile force: 150 kN. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Mode Number**} & \multicolumn{3}{c}{**50 kN**} \\ \cline{2-4}  & \(f_{r}\) **(Hz)** & \(f_{a}\) **(Hz)** & \(f_{r}\)-\(f_{a}\)/\(f_{r}\) **(\%)** \\ \hline
1 & 26.373 & 26.367 & 0.023 \\
2 & 73.795 & 73.792 & 0.004 \\
3 & 97.024 & 97.046 & 0.023 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Natural frequencies are experimentally determined starting from interferometric radar data (\(f_{r}\)) and accelerometric data (\(f_{a}\)). Applied tensile force: 50 kN.