# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 3 Method Evaluation Using Simulated Data
### Simulated Test  
The object of the simulated test was to assess analysis techniques for determining the EO and amplitude. Several exhaustive tests were performed using BTT data obtained from the simulator to evaluate the accuracy of the reconstruction method and their sensitivity to various test parameters. These parameters were:  
1. Probe spacing on the resonance (PSR);
2. The number of revolution (\(n\));
3. Engine order (EO);
4. Inter-blade coupling of the mistuned blade (ICoMB); \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \(\Delta f_{1}\) & \(\Delta f_{2}\) & \(\Delta f_{3}\) & \(\Delta f_{4}\) & \(\Delta f_{5}\) & \(\Delta f_{6}\) & \(\Delta f_{7}\) & \(\Delta f_{8}\) & \(\Delta f_{9}\) & \(\Delta f_{10}\) & \(\Delta f_{11}\) & \(\Delta f_{12}\) \\ \hline \(-\)0.706 & 3.165 & \(-\)5.328 & \(-\)9.319 & \(-\)5.796 & 1.334 & 1.565 & 1.806 & \(-\)0.521 & 0.734 & \(-\)1.904 & 3.448 \\ \hline \end{tabular}
\end{table} 
Table 1: The distribution of the mistuning coefficient \(\Delta f_{i}\) (%). Figure 4: The mistuning frequency of the blade. The number of probes (_g_);  
Noise-to-signal ratio (NSR). The NSR is defined as the ratio of the r.m.s. value of the noise to the value of the 'clean' amplitude of the blade. The NSR was set to 100 points averagely distributed in the range from 0.01 to 0.3. For the same NSR, a white noise generator was used to create 100 distinct noise sequences with which to corrupt the data obtained from the simulator. Each test was repeated using all the white noise sequences for each method on each NSR value. When there is noise in the data, the reconstruction method will contain two types of error [39]:  
1. Bias is where the estimates are incorrect. It is caused by a summated squared noise term in the \(\mathbf{B}^{\mathsf{T}}\mathbf{B}\) matrix that does not tend to zero but increases with increasing noise levels. This is an effect of the ICFF model that is used. 2. Scatter, where there is a range of estimates spread about the mean value, is caused by random variations in the data. The mean and variance of the recovered EO and amplitude were computed, from which 95 percent confidence intervals were obtained. The best method will exhibit the lowest bias and, preferably, lowest scatter. Six test cases were designed to analyze and compare the performance of the TCFF and ICFF under different test conditions. These are shown in Table 2. Test 1 was specifically chosen to be a very difficult case to analyze. Hence, the other tests were used to investigate whether specific changes in the parameters improve or degrade the performance of the method. The PSR is a key factor that affects the performance of all BTT algorithms. It is also commonly used to determine the quality of probe distribution for blade synchronous vibration measurement. Currently, there are corresponding methods for optimizing the probe installation layout [40]. Combined with the FEM, the optimization of probe layout can be achieved in actual engineering measurement. Therefore, we set the PSR to a high value (98%) in Test 1. In engineering applications, the probe is inevitably affected by the temperature, airflow, etc., and in severe cases, the measurement data will not be available. In this case, where a limited number of probes were used to measure blade vibration, the failed probe could not participate in the blade vibration analysis, which destroyed the optimization of the probe layout and resulted in a decrease of the PSR. For this reason, the PSR was set to a low value (55 percent) in Test 2. For TCFF, the blade vibration parameters can be calculated based on the single-revolution data obtained by multiple probes (usually four). However, it is impractical that the single-revolution data are used to identify the blade vibration parameters in engineering applications, considering the effects of system measurement errors.