## 2 Constant-Speed Blade Synchronous Vibration Parameter Recognition Based on BTC
\begin{array}{c}h_{0}=R-\sqrt{\left(e+r_{i}\cos(\theta_{0}\pm\alpha_ {i})\right)^{2}+\left(r_{i}\sin(\theta_{0}\pm\alpha_{i})\right)^{2}}\\ h_{1}=R-\sqrt{\left(e+r_{i}\cos(\theta_{1}\pm\alpha_{i})\right)^{2}+\left(r_{ i}\sin(\theta_{1}\pm\alpha_{i})\right)^{2}}\\ \vdots\\ h_{n-1}=R-\sqrt{\left(e+r_{i}\cos(\theta_{n-1}\pm\alpha_{i})\right)^{2}+\left( r_{i}\sin(\theta_{n-1}\pm\alpha_{i})\right)^{2}}\end{array} \right. \tag{4}\]  
Figure 1: Geometric model between BTC and BTVD, (**a**) the blade tip arrives early. (**b**) The blade tip arrives late. In the formula, \(R\) is the distance between the BTC-measuring sensor probe and the rotation axis and \(e\) is the radius of the blade disk. Substituting Equation (4) into Equation (3), we obtain (+\(\alpha_{i}\) for early arrival of blade tips and \(-\alpha_{i}\) for late arrival of blade tips):  
\[\left\{ \begin{array}{l}y_{0}=r_{i}\bigg{[}\arccos(\frac{(\frac{R-h_{0}}{2})^{ -2}-r_{t}^{2}}{2r_{i}})\pm\alpha_{i}\bigg{]}\\ y_{1}=r_{i}\bigg{[}\arccos(\frac{(\frac{R-h_{1}}{2})^{-2}-r_{t}^{2}}{2r_{i}}) \pm\alpha_{i}\bigg{]}\\ \vdots\\ y_{n-1}=r_{i}\bigg{[}\arccos(\frac{(\frac{R-h_{n-1}}{2})^{-2}-r_{t}^{2}}{2r_{ i}})\pm\alpha_{i}\bigg{]}\end{array} \right. \tag{5}\]  
Equation (5) converts the BTC measurement data into the BTVD data. The displacement constant values of synchronous vibration of constant-speed blades obtained by different sensors are different, and the displacement expression is  
\[y_{j}=A\sin(N_{e}\Omega t_{j}+\phi)+C\ \ (j=0,1,\ldots,n-1), \tag{6}\]  
Let the time of a blade passing through the Tip0 sensor be \(t_{0}=0\), and the time of the blade passing through the other sensors be \(t_{j}=\frac{\beta_{j}+2\pi k}{\Omega}\). Where \(k\) is the number of rotations. Substituting it into Formula (6), we can obtain:  
\[y_{j}=A\sin(N_{e}\beta_{j}+\phi)+C, \tag{7}\]  
The expansion of (7) can be written as follows:  
\[y_{j}=A\sin(N_{e}\beta_{j})\cos\phi+A\cos(N_{e}\beta_{j})\sin\phi+C, \tag{8}\]  
Then, \(j=0,j=1,j=2,\ldots,j=n-1\) are substituted into Formula (8), respectively, and we can obtain:  
\[\left\{ \begin{array}{l}y_{0}=A\sin\phi+C\\ y_{1}=A\cos(N_{e}\beta_{1})\sin\phi+A\sin(N_{e}\beta_{1})\cos\phi+C\\ y_{2}=A\cos(N_{e}\beta_{2})\sin\phi+A\sin(N_{e}\beta_{2})\cos\phi+C\\ \vdots\\ y_{n-1}=A\cos(N_{e}\beta_{n-1})\sin\phi+A\sin(N_{e}\beta_{n-1})\cos\phi+C\end{array}  \right., \tag{9}\]  
Divide Equation (9) into matrix expression \(Y=BX\), where the matrix forms of Y, B and X are:  
\[Y=(y_{0}\quad y_{1}\quad y_{2}\quad\ldots\quad y_{n-2}\quad y_{n-1})^{T}, \tag{10}\]  
\[B=\left[ \begin{array}{cccc}1&0&1\\ \cos(N_{e}\beta_{1})&\sin(N_{e}\beta_{1})&1\\ \cos(N_{e}\beta_{2})&\sin(N_{e}\phi_{2})&1\\ \vdots&\vdots&\vdots\\ \sin(N_{e}\beta_{n-1})&\sin(N_{e}\beta_{n-1})&1\end{array} \right], \tag{11}\]  
\[X=\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\end{array} \right]=\left[ \begin{array}{c}A\sin\phi\\ A\cos\phi\\ C\end{array} \right], \tag{12}\]  
From the matrix form of \(B\) and \(X\), it can be seen that the matrix \(B\) contains an unknown parameter \(Ne\), and the \(X\) vector contains three unknown parameters \(A\), \(\phi\) and \(C\), with a total of four unknowns. Therefore, at least four eddy current sensors are needed to effectively solve the blade vibration parameters. The vibration frequency doubling \(Ne\) is a positive integer, and the frequency-doubling value of the vibration of the rotating blade is in a certain range. The frequency-doubling traversal is carried out. All the _Ne_ values in the _Ne_ range are substituted into the \(Y=BX\), and the solution vector \(X_{k}\) is obtained by the least squares method [36]. \[X_{k}=\left(B^{T}B\right)^{-1}B^{T}Y, \tag{13}\]  
Substitute \(X_{k}\) into \(Y=BX\) and define their residual \(E_{k}\) with the actual vibration displacement \(Y\):  
\[E_{k}=BX_{k}-Y, \tag{14}\]  
In the formula, \(E_{k}=(e_{k0}\ e_{k1}\ e_{k2}\ldots e_{kn-1})^{T}\).