## 1 Introduction
According to the difference between the capacitance value and the electrical signal conversion method, this method can be divided into the DC method, the FM method and the AM method, and the highest accuracy can reach 15 \(\upmu\)m.  
The fourth method, the eddy current method, is based on the working principle of electromagnetic induction. When the measured blade passes through the excitation magnetic field, the magnetic flux changes, and the eddy current effect occurs in the blade. The induction coil in the sensor generates an electrical signal, and then the generated electrical signal output corresponds. The measurement of the BTC can be achieved via calibration to detect changes in electrical signals. According to the method of generating an excitation magnetic field, this method can be divided into the passive method and the active method, and the highest accuracy can reach 10 \(\upmu\)m.  
The fifth method, the microwave measurement method, is based on the working principle of short-range millimeter-wave ranging radar. The sensor installed in the casing sends out a microwave signal while receiving the reflected signal of the blade end face, and it realizes the BTC measurement by detecting the change in the signal frequency or amplitude. This method mainly includes the resonance frequency ranging method and the phase difference ranging method according to different measurement principles, and the highest accuracy can reach 25 \(\upmu\)m.  
The BTT is used to obtain the vibration parameters by processing the BTVD, but there are many uncertainties in the acquisition of the BTVD. One of the main uncertainties is the stable motion of the blade (i.e., the change in its average position and direction). The blade is deformed by the force during rotation [25; 26; 27], which affects the BTVD [28; 29; 30; 31] and the BTC. Since this paper mainly studies the constant-speed synchronous vibration of a low-pressure compressor (low temperature) blade, only the constant deformation and vibration deformation of the blade under aerodynamic force and the deformation of the rotor under centrifugal force are considered. The BTVD is usually obtained by processing the time series, and the time series of the tip arrival is obtained by the BTT sensor. But in this paper, the BTVD is obtained by using the BTC to BTVD conversion method. The sensor obtains the BTC, rather than a time series of blade tip arrivals. Purpose of the research: The modeling of the relationship between BTC and BTVD for vibration parameter identification by BTC. A new method for synchronous vibration parameter identification of constant-speed rotating blades based on BTC measurement is proposed. By modeling the geometric relationship between the BTC and the BTVD, the BTC is associated with the BTVD. BTC was obtained using the existing BTC-monitoring methods. The BTVD was obtained by using the geometric model. Finally, the nonlinear least squares method is used to identify the synchronous vibration parameters of the rotating blade under constant-speed conditions. This paper is organized as follows: the first part briefly introduces the BTC and its measurement method. The second part proposes a model for the geometric relationship between BTC and BTVD. The third part verifies the recognition accuracy of the method through numerical simulation. The fourth part verifies the feasibility of the method through experiments. Finally, some main conclusions are summarized in the fifth part.