# An Improved Circumferential Fourier Fit (CFF) Method for Blade Tip Timing Measurements
## 3 Method Evaluation Using Simulated Data  
### Simulated Model  
For the actual blade vibration measurement, even if the blade dynamic behavior is fully understood, it is still difficult to eliminate uncertainties in the measurement data [33; 34], such as speed fluctuations, system random errors, etc. These uncertainties will affect the evaluation accuracy, therefore, it is better to use simulated data without measurement uncertainties to evaluate the accuracy of the reconstruction methods. A 12-blade assembly simulator was used to evaluate the ICCF performance. The schematic of the simulated model was shown in Figure 3. A mass-spring-damper system represents each blade, which is coupled to its two neighbors through two further spring-damper assemblies. A mathematical model was employed to simulate the forced vibration of the rotating bladed assembly. This mathematical model ignored the effects of the centrifugal force and temperature on the blade stiffness. The mathematical model of the bladed assembly is given by:  
\[\mathbf{M}\ddot{y}+\mathbf{C}\dot{y}+\mathbf{K}y\ =\ \mathbf{F}(t) \tag{13}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), \(\mathbf{K}\), and \(\mathbf{F}(t)\) represent the mass matrix, damping matrix, stiffness matrix, and excitation force vector, respectively. Let \(\mathbf{W}(i,j)\) (\(1\leq i\leq 12\) and \(1\leq j\leq 12\)) represents the element at the \(i\)th row and \(j\)th column of the matrix \(\mathbf{W}\), and above matrixes can be expressed as:  
\[\mathbf{M}(i,j)\ =\ \left\{ \begin{array}{ll}m_{i}&i\ =\ j\\ 0&\mbox{\it others}\end{array} \right. \tag{14}\] \[\mathbf{C}(i,j)\ =\ \left\{ \begin{array}{ll}c_{i,j-1}+c_{j}+c_{i,j+1}&i =\ j\\ -c_{i,j}&\left|i-j\right|\ =\ 1\\ 0&\mbox{\it others}\end{array} \right.\] \[\mathbf{K}(i,j)\ =\ \left\{ \begin{array}{ll}k_{i,j-1}+k_{j}+k_{i,j+1}&i =\ j\\ -k_{i,j}&\left|i-j\right|\ =\ 1\\ 0&\mbox{\it others}\end{array} \right.\]  
In Equation (14), use 1 for subscripts greater than 12 and 12 for subscripts less than 1. The excitation force \(f_{i}\) (t) on blade \(i\) is given by:  
\[f_{i}(t)\ =\ F_{i}\sin(\mathrm{EO}\omega_{v}t+\frac{2\pi\mathrm{EO}}{N}i),\mbox{ \it for}\ \ \ i\ =\ 0,1,\ldots N \tag{15}\]  
where \(F_{i}\) is the amplitude of the excitation force, and \(N\) represents the count of the blade. \(N\) was set to 12 in this paper. Figure 3: The simulated model. \(m_{i}\), \(k_{i}\), and \(c_{i}\) (\(i=1,2,3\)... 12) represent the mass, stiffness, and damping coefficient of blade \(i\), respectively. \(k_{i,j}\) and \(c_{i,j}\) (\(j-j\)\(|=1\)) represent the coupling stiffness and coupling damping coefficient between adjacent blades. The quality of probe distribution is represented by the probe spacing on the resonance (PSR). The PSR is the ratio of the difference between the times of arrival of a blade at the first and last probe to the period of the blade response at resonance [35], i.e.,:  
\[\text{PSR }=\omega_{n}(T\alpha A_{\text{last}}-T\alpha A_{\text{first}})/(2\pi), \tag{16}\]  
where \(\omega_{n}\) represents the natural angular frequency. Further details on the mathematics of the simulator are given by the literature [36]. According to the literature [37; 38], the mistuning coefficient \(\Delta f_{i}\) and the coupling coefficient \(h_{i}\) are defined to characterize the degree of blade mistuning and the coupling between blades. The concrete expressions of these two parameters are:  
\[ \begin{array}{l}\Delta f_{i}\ =\ \frac{k_{i}}{k}-1\\ h_{i}^{2}\ =\ \frac{k_{i,j}}{k_{i}}\ \,\ \ \ \ \ \left|i-j\right|\ =\ 1\end{array}  \tag{17}\]  
where the subscripts \(i\) and \(j\) represent the blade number, and \(k\) represents the blade nominal stiffness. Let the nominal mass \(m\), stiffness \(k\), and damping \(c\) be 1 kg, \(8.1\times 10^{5}\) N/m, and 9 N's/m, respectively.