## 5 Conclusions  
In this paper, a new method for the identification of synchronous vibration parameters of constant-speed rotating blades based on BTC measurement is proposed. By establishing a geometric model between BTC and BTVD, BTC is associated with BTVD. Through the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sensors** & **Tip0** & **Tip1** & **Tip2** & **Tip3** \\ \hline Measured value & 0.19 & 0.21 & 0.21 & 0.20 \\ (mm) & (early arrival) & (late arrival) & (early arrival) & (early arrival) \\ \hline \hline \end{tabular}
\end{table}
Table 16: Measured BTC values (mm).  
Figure 21: Vertical axis integrated blade-monitoring station.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Vibration Parameters** & \(N_{e}\) & \(A\) & \(\varphi\) & \(C\) \\ \hline Recognition results & 8 & 2.5907 & \(-1.5487\) & \(-0.1453\) \\ \hline \hline \end{tabular}
\end{table}
Table 17: Recognition results of blade tip parameters.  
existing BTC-monitoring method, the BTC of the blade tip passing through each sensor is obtained. The BTT method is used to determine whether the blade tip arrives in advance or with lag, and then the geometric model is used to obtain BTVD. Finally, the nonlinear least squares method is used to identify the synchronous vibration parameters of the constant-speed rotating blade. And the identification accuracy of the method is verified by numerical simulation. The conclusions are described as follows: with an increase in amplitude, the higher the accuracy of the vibration parameter identification proposed in this paper; with a decrease in the random error of the BTC measurement, the higher the accuracy of the vibration parameter identification proposed in this paper; with a decrease in the random error in the measurement of the blade disk dimensions, the higher the accuracy of the vibration parameter identification proposed in this paper; the smaller the ratio of blade length to the disk radius, the higher the accuracy of the vibration parameter identification method introduced in this paper. Since the compressor and turbine blade disk structures have a small \(r/e\) ratio, the method proposed in this paper is suitable for the identification of synchronous vibration parameters of compressor blades and turbine blades. In the future, we will analyze the uncertain factors such as temperature and centrifugal force measurement at different speeds and carry out statistical analysis to gradually improve this method. With the development of the BTC measurement method and our research, the recognition accuracy of this method will also be improved. This method will be applicable to identification of all rotating blade vibration parameters.  
Conceptualization, L.Z.; methodology L.Z.; software, Y.X.; validation, L.Z.; formal analysis, Y.X.; investigation, C.C.; data curation, Y.X.; writing--original draft preparation, L.Z.; writing--review and editing, L.Z., Y.X. and J.C.; visualization, Q.S.; supervision, L.Z.; project administration, L.Z. All authors have read and agreed to the published version of the manuscript.  
The work is supported by National Natural Science Foundation of China (Grant No. 51505206), Liaoning Province Applied Basic Research Program Project (Grant No. 2023H2/101300227), Key Laboratory of Vibration and Control of Aero-Propulsion System, Ministry of Education, Northeastern University (Grant No. VCAME202211), Liaoning Provincial Natural Science Foundation Guidance Program Project (Grant No. 2019-ZD-0694), Liaoning Provincial Department of Education Scientific Research General Project (Grant No. L2014246) and Liaoning University of Technology Teacher Research Start Fund Project (Grant No. X201202).  
Not applicable.  
The data used to support the findings of this paper are available from the corresponding author upon request. The data are not publicly available due to privacy.  
The authors declared no conflicts of interest.