# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 4 Method Evaluation Using Experimental Data  
The experimental data that was provided by the author of the literature [36] was used to further verify the feasibility of the reconstruction method proposed in this paper. The test rig in the literature [36] was shown in Figure 11a. The radius of the rotor is 60 mm, and there are eight blades. The once per revolution (OPR) sensor was installed near the shaft to provide the rotating speed and a timing reference through detecting a marking on the shaft. Seven optic probes were installed on the casing to measure the blade vibration. The installation angles of the probes were 0\({}^{\circ}\) (P\({}_{0}\)), 18.56\({}^{\circ}\) (P\({}_{1}\)), 36.17\({}^{\circ}\) (P\({}_{2}\)), 53.75\({}^{\circ}\) (P\({}_{3}\)), 72.42\({}^{\circ}\) (P\({}_{4}\)), 119.82\({}^{\circ}\) (P\({}_{5}\)), and 239.06\({}^{\circ}\) (P\({}_{6}\)). The nitrogen shock was used as an exciting force to the rotating blades. The mode frequency was plotted on the vertical axis in the Campbell diagram of Figure 11b. The diagram plotted the resonant frequency on the vertical axis and the rotating speed on the horizontal axis. The EO lines showed a range from 9 to 15 (blue lines). The seven black dots in the Campbell diagram represent the intersection of the EO lines with the mode frequency. The nominal resonant speeds corresponding to these seven intersections are 7081 rpm (EO = 15), 7613 rpm (EO = 14),  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**NSR**} & \multicolumn{2}{c}{**Test 1**} & \multicolumn{2}{c}{**Test 2**} & \multicolumn{2}{c}{**Test 3**} & \multicolumn{2}{c}{**Test 4**} & \multicolumn{2}{c}{**Test 5**} & \multicolumn{2}{c}{**Test 6**} \\ \cline{2-13}  & **TCFF** & **ICFF** & **TCFF** & **ICFF** & **ICFF** & **ICFF** & **ICFF** & **ICFF** & **ICFF** & **ICFF** & **ICFF** \\ \hline
0.01 & 13.49\% & 4.06\% & — & — & 28.74\% & 23.82\% & 29.41\% & 20.04\% & 15.97\% & 14.53\% & 18.29\% & 10.32\% \\
0.05 & 13.42\% & 4.00\% & — & — & 28.59\% & 23.64\% & 29.29\% & 19.96\% & 15.87\% & 14.46\% & 18.19\% & 10.19\% \\
0.10 & 13.03\% & 3.56\% & — & — & 28.29\% & 23.37\% & 28.70\% & 18.97\% & 15.59\% & 14.25\% & 17.97\% & 10.02\% \\
0.15 & 12.69\% & 3.35\% & 18.63\% & 19.07\% & 27.63\% & 22.48\% & 27.62\% & 17.53\% & 15.72\% & 14.47\% & 17.38\% & 9.30\% \\
0.20 & 12.23\% & 2.98\% & 19.10\% & 19.38\% & 26.83\% & 21.63\% & 26.39\% & 15.25\% & 15.24\% & 14.16\% & 16.58\% & 8.47\% \\
0.25 & 11.30\% & 1.92\% & 18.30\% & 19.50\% & 26.22\% & 20.94\% & 24.54\% &8202 rpm (EO = 13), 8907 rpm (EO = 12), 9724 rpm (EO = 11), 10,710 rpm (EO = 10), and 11,890 rpm (EO = 9), respectively. The displacement of blade 7 measured by probe P\({}_{0}\) for rotating speed 6500-12,100 rpm was shown in Figure 12. In the whole speed up, there were seven resonances excited by the nitrogen shock, although the vibration intensity was not completely the same. The EO corresponding to each resonance was predicted through the Campbell diagram (Figure 11b), which was labeled in Figure 12. The similar resonances were also measured by other probes, and here it did not show these vibration waveforms one by one. In the engineering applications, the EO can be predicted through the Campbell diagram, like the experiment in this paper. The EO identification step can be skipped on the condition that the true EO had been known before parameter identification. In this case, the correct EO would be used directly by identification methods to calculate the amplitude, phase, and direct current offset. In this section, the other parameters would be calculated based on the EO that was from prediction rather than identification. Therefore, the reconstruction accuracy was compared for each method base on the EO known. Although the EO corresponding to each resonance can be predicted through the Campbell diagram, the corresponding amplitude can not be known exactly, which was not the same as the simulated  
Figure 11: (**a**) The test rig and (**b**) the blade Campbell diagram.