# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 3 Method Evaluation Using Simulated Data
### Simulated Test
Just as the AR method does [39], it is reasonable to select multi-revolution data, including the blade synchronous resonance region, to determine the true vibration parameters based on the average of the calculation results. For any curve fitting algorithm, such as the single-parameter method, the amount of selected data should include the complete blade resonance response region as much as possible, although there are no specific rules on how many data points should be selected. In the simulated tests, 30 (Test 1) and 50 (Test 3) data points were selected \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test** & **PSR** & \(n\) & **True EO** & \(g\) & **ICoMB** \\ \hline Test 1 & 98\% & 30 & 10 & 4 & No \\ Test 2 & 55\% & 30 & 10 & 4 & No \\ Test 3 & 98\% & 50 & 10 & 4 & No \\ Test 4 & 98\% & 30 & 13 & 4 & No \\ Test 5 & 98\% & 30 & 10 & 2 & No \\ Test 6 & 98\% & 30 & 10 & 4 & Yes \\ \hline \hline \end{tabular}  
* The parameters whose background was shaded were the changed parameters relative to Test 1.  
\end{table} 
Table 2: The simulated test. for identifying the vibration parameters to demonstrate the effects of the data amounts on the quality of the identification of each method. BTT technology is sensitive to a mode with a large amplitude at the blade tip, such as the first-order bending mode. In actual rotating machinery operation, due to mechanical structure or airflow disturbance, the same vibration mode of the blade may be excited at different resonance rotating speeds corresponding to different EO. For BTT algorithms, analysis of the blade vibration at different resonance rotating speeds is essentially a process of reconstructing signals with different degrees of under-sampling. The EO (Test 1: EO = 10; Test 4: EO = 13) was used to analyze and compare the sensitivity of TCFF and ICFF to different degrees of under-sampling, and the EO was also an important parameter for the blade vibration analysis. Based on the CFF theory, at least three probes are needed to monitor blade synchronous vibration. The least-square method is used to calculate the blade synchronous vibration parameters if more than three probes are used. The TCFF usually uses four probes to increase robustness. However, [PERSON] used seven probes in the experimental verification of TCFF [31]. So far, there is no optimal choice for the number of probes. Test 5 was designed to verify the performance of TCFF and ICFF on the different number of probes. Mistuning is a universal feature of the objective existence of the blades, and if inter-blade coupling of the mistuned blade exists, it will affect the performance of the BTT algorithms that only analyze single-frequency signals, such as the single-parameter method, the two-parameter plot method, CFF, and TCFF. The same applies to the ICFF method proposed in this paper. Therefore, Test 6 was specially designed to analyze and compare the sensitivity of TCFF and ICFF to ICoMB. The noise in the measurement data is difficult to eliminate. The \"clean\" simulated data was polluted by noise in all simulated tests to make the data closer to the real measurements. The anti-noise performance of each method was verified by using simulated data of different pollution degrees that were represented by the NSR value. The monitor positions of the probes in Test 1, Test 3, and Test 6 were 0\({}^{\circ}\), 13.1\({}^{\circ}\), 25.3\({}^{\circ}\), and 35.3\({}^{\circ}\). The monitor positions of the probes in Test 4 were 0\({}^{\circ}\), 11.3\({}^{\circ}\), 13.3\({}^{\circ}\), and 25.3\({}^{\circ}\). In Test 2, the monitor positions of the probes were 0\({}^{\circ}\), 6.3\({}^{\circ}\), 10.1\({}^{\circ}\), and 19.8\({}^{\circ}\). In Test 5, the monitor positions of the probes were 0\({}^{\circ}\), 6.3\({}^{\circ}\), 13.1\({}^{\circ}\), 19.1\({}^{\circ}\), 25.3\({}^{\circ}\), 30.2\({}^{\circ}\), and 35.3\({}^{\circ}\). The PSR in Table 2 was calculated according to Equation (16).