# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 2 The Improved Circumferential Fourier Fit (ICFF) Method
### The Derivation of ICFF  
Equation (5) is transformed by triangle formulas:  
\[y\ =\ A\sin(EO\overline{\beta})\cos(\varphi)+A\cos(EO\overline{\beta})\sin( \varphi)+d \tag{6}\]  
TCEF usually needs at least four probes to monitor blade vibration. In this section, assuming that there are \(g\) probes, the blade vibration displacement sequence can be represented according to Equation (6):  
\[\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{g}\end{array} \right]=\left[ \begin{array}{ccc}\sin(EO\overline{\beta}_{1} )&\cos(EO\overline{\beta}_{1})&1\\ \sin(EO\overline{\beta}_{2})&\cos(EO\overline{\beta}_{2})&1\\ \vdots&\vdots&\vdots\\ \sin(EO\overline{\beta}_{g})&\cos(EO\overline{\beta}_{g})&1\end{array} \right] \left[ \begin{array}{c}A\cos(\varphi)\\ A\sin(\varphi)\\ d\end{array} \right] \tag{7}\]  
That is:  
\[\mathbf{Y}\ =\ \mathbf{B}\mathbf{X} \tag{8}\]  
Figure 2: The schematic of the blade tip timing (BTT) measurement. The once per revolution (OPR) sensor monitors the rotating speed through the mark on the shaft and provides a timing reference for the BTT measurement system. The probe is responsible for monitoring the ToA. \(R\) represents the distance from the blade tip to the center of the rotor, and \(y\)/\(R\) represents the circumferential angle of the blade vibration. \(\beta\) represents the monitoring position, and \(\varepsilon\) represents the angle of the blade from the first probe when the mark passes the OPR sensor. where **Y** is the displacement vector, **B** is the coefficient matrix, and **X** is the parameter matrix. The main difference between ICFF and TCFF is the constituent parameters of the coefficient matrix **B**, although Equation (7) in this paper is the same in form as TCFF. For the TCFF method, the coefficient matrix **B** is completely determined by the probe mounted position \(\beta\) and the initial angle \(\varepsilon\), that is, once the probe installation scheme is determined, the coefficient matrix **B** is also determined. This means that the coefficient matrix **B** is a constant in the TCFF method. The same applies to the CFF method. However, the coefficient matrix **B** in ICFF is a function of three parameters: the probe installation angle \(\beta\), the initial angle \(\varepsilon\), and the blade vibration displacement \(y\)/\(R\). According to the derivation in Section 2.2, blade vibration displacement is also one of the most important parameters affecting the ToA. In this case, the coefficient matrix **B** is no longer a constant but can be adjusted in real-time according to the actual vibration displacement of the blade. Based on the principle of least squares and EO traversal, when the possible engine order EO\({}_{k}\) (\(k\) = 1, 2, 3...) is selected, there is:  
\[\textbf{X}_{k}\ =\ \left(\textbf{B}_{k}^{T}\textbf{B}_{k}\right)^{-1}\textbf{B}_{ k}^{T}\textbf{Y}. \tag{9}\]  
The condition number is the indicator of the quality of the matrix **B**, which influence the accuracy of **X** directly. The larger the condition number, the more ill-condition matrix **B** is, and the more sensitive of the calculated results are to the measurement error. An effective way to improve the quality of the matrix **B** is to adjust the sensor layout. Therefore, based on the minimization the condition number of matrix **B**, a method about determining the sensor layout can be derived. The condition number mentioned above and the probe spacing of the resonance (PSR) used later in the paper as an indicator of sensor distribution both are methods essentially to determine how the sampled interval of periodic signals are arranged. However, PSR is more common.