# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 3 Method Evaluation Using Simulated Data
### Simulated Test
If the option of the ICoMB is \"NO\", it means that the coupling coefficient is set to zero; if it is \"Yes\", it means the coupling coefficient is set to nonzero. In this paper, the coupling coefficient \(h_{i}\) (\(i\) = 1, 2, 3..., 12) was set to 0.1 for the option \"Yes\". It is must be stressed that the blade vibration does not influence the adjacent blades when the coupling coefficient is set to zero, even under the condition that the blade is mistuned. For this, it is still a single-frequency vibration for the single blade. However, the synchronous resonance of the single blade will be a superposition of multiple-frequency vibration under the condition that the coupling coefficient is set to nonzero, and it is difficult to distinguish. The coupled vibration of the mistuned blade is a serious challenge for TCFF, since it assumes the blade response is of single-frequency vibration. According to the results shown in Figure 4, the natural frequency of blade 7 is close to that of the adjacent blades. To evaluate the performance of ICFF under the condition of blade mistuning and inter-blade coupling, the simulated data of blade 7 were selected in the simulated tests. The rotating speed was set to accelerate uniformly from 300 to 2400 rpm, which included the resonance rotating speed for all simulated tests. Take the resonance rotating speed as the reference and select half \(n\) points before and after the reference point to compose the number of revolutions \(n\). The results of each simulated test were analyzed as follows. \(\bullet\) Test 1:  
As expected, Test 1 was a very difficult case. As shown in Figure 4(a), the EO identification started to produce biased results at about NSR 0.16 for both methods. For tests with correct EO identification, the 95% confidence intervals of the relative error of the amplitude identification were calculated, which was shown in Figure 5b. The intervals increased with the increase of the NSR for both methods. However, the relative error of amplitude identification of ICFF was lower than that of TCFF. \(\bullet\) Test 2:  
Reducing the PSR value to 55 percent produced the most dramatic degrade in the quality of the identification. The correct EO identification did not occur for ICFF before NSR 0.15, and the deviation of EO identification was very large for both methods compared with Test 1. In order to compare the two methods synchronously, Figure 6a only showed that the results of the EO identification in the NSR range of 0.15 and 0.30. The relative error of amplitude identification was calculated after NSR 0.15. The scatter of the relative error was higher than that of Test 1. As already discussed by other literature [35; 36; 39], curve fits of lower PSR data are generally not conducive to parameter identification. Figure 5: Results of Test 1. (**a**) 95% confidence intervals of the engine order (EO) identification and (**b**) 95% confidence intervals of the relative error of the amplitude identification. Figure 6: Results of Test 2. (**a**) 95% confidence intervals of the EO identification and (**b**) 95% confidence intervals of the relative error of the amplitude identification. \(\bullet\) Test 3:  
Increasing the number of revolutions over which data were used for the identification process can improve the results. The results of EO identification started to be biased when the NSR increase to about 0.18 for both methods, which was shown in Figure 7a. The NSR corresponding to unbiased identification was higher 0.03 than Test 1. However, relative error for amplitude identification increased with increasing the number of data, which was caused by a summated squared noise term in the \(\mathbf{B}^{\text{T}}\mathbf{B}\) matrix. This is an effect of the method model that is used. Although the relative error was increased for both methods, the ICFF error increased less than the TCFF. \(\bullet\) Test 4:  
The results of Test 4 were shown in Figure 8. Increasing the EO of the tip timing data had no visible effect on the accuracy of the EO identification for ICFF.