# Vibration Parameters Identification of Rotating Blades Based on Blade Tip-Timing Sensor Waveforms
## 2 Waveform Mapping Theory
### Identification of Blade Vibration Parameters Based on Waveform Mapping Method  
The single harmonic vibration equation of blade is as follows:  
\[\Delta x(t)=A_{0}\sin(N_{e}\Omega t+\varphi)+C, \tag{4}\]  
where \(\Delta x(t)\) is the blade tip vibration displacement, \(A_{0}\) is the vibration amplitude of blade tip, \(N_{e}\) is the excitation order, \(\Omega\) is the rotational speed, \(\varphi\) is the vibration initial phase and \(C\) is the vibration constant offset. The position of the first blade root coincides with that of the OPR groove in the axial direction. When the first blade tip passes through the first tip-timing sensor, the circumferential displacement of the first blade tip relative to the OPR sensor is as follows:  
\[x(t)=A_{0}\sin(N_{e}\Omega t+\varphi)+C+R\Omega t, \tag{5}\]  
when the blade tip rotates into the sensor measurement range; that is, when \(x(t)\in[Ra_{1}-Ra_{\max},Ra_{1}+Ra_{\max}]\), \(R\) is the blade tip radius, \(\alpha_{1}\) is the angle between the first blade tip-timing sensor and the OPR sensor and \(\alpha_{\max}\) is the maximum angle between the blade center line and the sensor center line in the measurement range. According to the mapping relationship between the dynamic test voltage waveform and the static calibration voltage waveform of blade tip timing sensor, as shown in Figure 2, in the same monotone interval, the data points with the same voltage values of the two waveforms are in the same time state; that is:  
\[g(t)=U=f(x(t)). \tag{6}\]  
The inverse function of Equation (6) is solved as follows:  
\[x(t)=f^{-1}[g(t)]. \tag{7}\]  
Substituting Equations (4) and (5) into Equation (7), and sorting out the following results,  
\[\Delta x(t)=A_{0}\sin(N_{e}\Omega t+\varphi)+C=f^{-1}[g(t)]-R\Omega t. \tag{8}\]  
When the time of the first blade passing through the \(i\)-th data point of the first waveform of the first blade tip timing sensor is \(t_{i}\), then Equation (8) becomes:  
\[\Delta x(t_{i})=A_{0}\sin(N_{e}\Omega t_{i}+\varphi)+C=f^{-1}[g(t_{i})]-R \Omega t_{i},\;i=1,2,\cdots,n. \tag{9}\]Here, \(n\) is the number of data points in the waveform. Let  
\[Y=\left(f^{-1}[g(t_{1})]-R\Omega t_{1}\quad f^{-1}[g(t_{2})]-R\Omega t_{2}\quad \cdots\quad f^{-1}[g(t_{n-1})]-R\Omega t_{n-1}\quad f^{-1}[g(t_{n})]-R\Omega t_{n }\right)^{T} \tag{10}\]  
\[B=\left[ \begin{array}{ccc}\cos(N_{\epsilon}\Omega t_{1})&sin(N_{\epsilon} \Omega t_{1})&1\\ \cos(N_{\epsilon}\Omega t_{2})&sin(N_{\epsilon}\Omega t_{2})&1\\ \vdots&\vdots&\vdots\\ \cos(N_{\epsilon}\Omega t_{n-1})&sin(N_{\epsilon}\Omega t_{n-1})&1\\ \cos(N_{\epsilon}\Omega t_{n})&sin(N_{\epsilon}\Omega t_{n})&1\\ \end{array} \right] \tag{11}\]  
\[Z=\left[ \begin{array}{c}z_{1}\\ z_{2}\\ z_{3}\end{array} \right]=\left[ \begin{array}{c}A_{0}\sin\varphi\\ A_{0}\cos\varphi\\ C\end{array} \right] \tag{12}\]  
Equation (9) is expressed in matrix form:  
\[Y=BZ. \tag{13}\]  
The range of the main excitation order of a rotating machinery can be estimated, since the excitation order \(N_{\epsilon}\) is a natural number. We select a certain range of excitation order, substitute its value into Equation (13) and use the least square method to solve the corresponding solution vector. The solution vector is as follows:  
\[Z_{k}=\left(B^{T}B\right)^{-1}B^{T}Y. \tag{14}\]  
Equation (14) is substituted into Equation (13), and the measured vibration displacement value \(Y\) is subtracted, then the residual is  
\[E_{k}=BZ_{k}-Y, \tag{15}\]  
Figure 2: Mapping relationship between blade tip-timing waveform and static calibration waveform. (**a**) Waveform mapping when the blade is not vibrating. (**b**) Waveform mapping when the blade is vibrating. where \(E_{k}\) is a column vector of element \(e_{kj}\), \(j=1\), \(j=2\)...... \(j=n-1\).