## 2 Constant-Speed Blade Synchronous Vibration Parameter Recognition Based on BTC  
In this paper, a new method for identifying the synchronous vibration parameters of constant-speed rotating blades based on BTC measurement is proposed. When the rotating blade does not vibrate, the blade tip passes through each BTC measurement sensor, and the BTC measured by each sensor is basically the same. When the blade vibrates, the BTC of the blade tip passing through each BTC measurement sensor is no longer the same but varies periodically. The BTC of the blade tip passing through each sensor is measured by each BTC-measuring sensor, and then BTVD is obtained. By geometrical modeling between BTC and BTVD, the measurements of BTC are transformed into the measurements of BTVD, and then the synchronous vibration parameters of the isotropic rotating blade are identified using the nonlinear least squares method [32; 33; 34; 35]. The method used in this paper is the same as the BTT test scheme. The test system includes an infrared rotating phase sensor, eddy current BTC measurement sensor and the software and hardware of the signal acquisition, processing and analysis system. The eddy current sensors are mounted on the sensor bracket. The serial number of the sensor is set to 0, 1, 2, \(\ldots\), \(n-1\). The sensor is TIP\(j\) (\(j=0,1,\ldots,n-1\)). The installation angle of each sensor relative to the Tip0 sensor is \(\beta_{0}\), \(\beta_{1}\), \(\beta_{2}\), \(\ldots\)\(\beta_{n-1}\)(\(\beta_{0}=0\)). The installation angle of the sensor \(j\) relative to the Tip0 sensor is \(\beta_{j}\)(\(j=0,1,\ldots,n-1\)). \(\theta_{j}\) is the vibration angle of the blade tip when the blade tip passes through the TIP\(j\). \(r\) is the static blade. \(r_{i}\) is the deformed blade. \(R\) is the distance between the probe of the BTC measurement sensor and the rotation axis. \(e\) is the radius of the blade disk. The geometric model relationship between BTC and BTVD is shown in Figure 1 ((a) shows the geometric model of the relationship between BTC and BTVD when the blade tip arrives early. (b) is the geometric model of the relationship between BTC and BTVD when the blade tip arrives late. We will choose different models and formulas according to different blade arrival states). As shown in Figure 1, the length of the blade in the static state is \(r\). When the blade disk rotates at the speed \(\Omega_{i}\), the deformation of the blade tip in the x and y directions is \(\Delta x_{i}\) and \(\Delta y_{i}\), respectively. The length of the deformed blade is:  
\[r_{i}=\sqrt{\left(r+\Delta y_{i}\right)^{2}+\Delta x_{i}^{2}}, \tag{1}\]  
The deformation angle of the blade is:  
\[\alpha_{i}=\arctan(\frac{\Delta x_{i}}{r+\Delta y_{i}}), \tag{2}\]  
When the blade tip passes through the BTC-measuring sensors Tip0, Tip1 and Tip\(n-1\), the BTVD are \(y_{0}\), \(y_{1}\), \(\ldots\), \(y_{n-1}\), respectively. From the geometric relationship, it can be known that:  
\[\left\{ \begin{array}{c}y_{0}=r_{i}\theta_{0}\\ y_{1}=r_{i}\theta_{1}\\ \vdots\\ y_{n-1}=r_{i}\theta_{n-1}\end{array} \right., \tag{3}\]  
In the formula: \(\theta_{0}\), \(\theta_{1}\), \(\ldots\), \(\theta_{n-1}\) are the blade tip vibration angles when the blade tip passes through the BTC measurement sensors Tip0, Tip1, \(\ldots\), Tip\(n-1\), respectively. According to the geometric relationship, the BTC values measured when the blade tip passes through the BTC-measuring sensors Tip0, Tip1 and Tip\(n-1\) are \(h_{0}\), \(h_{1}\), \(\ldots\) and \(h_{n-1}\), respectively, and satisfy the following formula (\(+\alpha_{i}\) for early arrival of blade tips and \(-\alpha_{i}\) for late arrival of blade tips):  
\[\left\{