# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 4 Method Evaluation Using Experimental Data
Figure 12: The displacement of blade 7 measured by probe P\({}_{0}\). tests where the true signal was obtained in advance. Even if the strain gauges were applied, it also needs a very complex technique for converting the strain to the amplitude of the blade tip, and the strain measurement error as an integrated part of strain data was also transmitted to the amplitude. Therefore, where the true amplitude is unknown, the relative error of reconstruction (REoR) was used to evaluate the quality of reconstruction. If the REoR is greater than zero, the higher REoR, the higher quality of reconstruction for ICFF. In contrast, if the REoR is negative that means the reconstruction accuracy of ICFF is lower than TCFF. The REoR is given by:  
\[\text{REoR}=\frac{\text{STDoE}_{\text{ICFF}}-\text{STDoE}_{\text{ICFF}}}{ \text{STDoE}_{\text{ICFF}}}\times 100\% \tag{19}\]  
where the STDoE represents the standard deviation of reconstruction error for each method, which is indicated by the subscript. The STDoE is given by:  
\[\text{STDoE}=\frac{1}{n}\sum_{j=1}^{n}\sqrt{\frac{1}{g}\sum_{i=0}^{g-1}\left( y_{i,j}-y_{i,j}^{*}\right)^{2}}\times 100\% \tag{20}\]  
where \(n\) represents the number of revolutions, \(g\) represents the number of probes, \(y_{i,j}\) represents the measurement displacement by probe \(i\) at revolution \(j\), and \(y_{i,j}^{*}\) represents the reconstruction displacement. In this experiment, the EO has been determined based on the FEM in advance. So, the true EO can be directly used without identification, which like the CFF does. Based on the results of simulated tests, the more number of probes and fitted data, the higher the reconstruction error for amplitude calculated. Therefore, just four probes (P\({}_{0}\), P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\)) instead of all were selected to identify the amplitude. For each resonance analysis, the number of fitted data was 20. The reconstruction data of blade 7 measured by probe P\({}_{0}\) was plotted in Figure 13. Each resonance was divided by the blue lines. The reconstruction resonance waveforms did not display the whole since only 20 data points were selected for each resonance. The results of reconstruction for measured data by the other probe were similar to Figure 13, which were not shown one by one. It was difficult to determine the quality of the identification only through Figure 13. Therefore, the STDoE and REoR were calculated based on the reconstruction data to achieve the purpose of comparing the quality of the identification. The results of identification for each method were summarized in Table 5.  
The REoRs were all greater than zero for all resonance analysis, which was shown in Table 5. That means the reconstruction accuracy of ICFF was higher than that of TCFF. Besides, even for lower PSR like the EO 14 corresponding to PSR 9.03 percent, as long as the true EO is known, the results of amplitude identification still was credible for both methods. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & & & \\ \cline{3-10}
**Resonance** & **PSR** & **EO** & **Amplitude (\(\mu\)m)** & **Direct Current Offset (\(\mu\)m)** & **STDoE (\(\mu\)m)** & **REoR** \\ \cline{3-10}  & & & **TCEF** & **ICFF** & **ICFF** & **ICFF** & **ICFF** & **ICFF** & \\ \hline
1 & 23.96\% & 15 & 63.01 & 43.01 & 4.84 & 3.83 & 11.51 & 9.69 & +15.81\% \\
2 & 9.03\% & 14 & 78.48 & 67.59 & 7.32 & 8.67 & 27.11 & 23.85 & +12.02\% \\
3 & 94.10\% & 13 & 83.55 & 71.19 & 13.50 & 12.70 & 18.48 & 16.72 & +9.52\% \\
4 & 79.17\% & 12 & 99.28 & 85.26 & 2.29 & 2.23 & 31.61 & 29.02 & +8.16\% \\
5 & 64.24\% & 11 & 88.47 & 75.75 & 3.38 & 3.77 & 16.20 & 14.99 & +7.47\% \\
6 & 49.31\% & 10 & 127.63 & 107.87 & 2.36 & 1.73 & 28.16 & 24.05 & +14.57\% \\
7 & 34.38\% & 9 & 84.92 & 75.03 & 4.69 & 3.77 & 10.31 & 9.28 & +9.99\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The results of the identification.