# Vibration Parameters Identification of Rotating Blades Based on Blade Tip-Timing Sensor Waveforms
## 3 Analysis of Influencing Factors of Blade Tip-Timing Waveform Shape  
The blade dynamics model is established to simulate the blade vibration. Assuming that the blade performs single-frequency simple harmonic vibration, the vibration equation is as in Equation (8), where the vibration amplitude \(A_{0}=3\) mm, excitation order \(N_{e}=8\), rotational speed \(\Omega=20\pi\) rad\(/\)s, initial phase \(\varphi=-\frac{\pi}{10}\) rad and constant offset \(C=0.2\) mm. The vibration displacement and vibration velocity waveform of the blade tip are shown in Figure 3. In Figure 3, Case 1 is the vibration equilibrium position; the vibration displacement is the smallest and the vibration velocity is the positive maximum. Case 2 is the position of the positive maximum value of the vibration displacement, and the vibration velocity passes the 0 point from positive to negative. Case 3 is the vibration equilibrium position; the vibration displacement is the smallest and the vibration velocity is the negative maximum. Case 4 is the position of the negative maximum value of the vibration displacement, and the vibration velocity passes the 0 point from negative to positive. Figure 4 shows the waveform of the blade tip passing through the timing sensor when the blade is rigidly rotated and vibrating. Figure 4a, Case 1, shows the positive synchronization of the vibration. Vibration displacement changes from negative to positive and passes through the equilibrium position, and the vibration displacement is close to 0, so the times when the highest points of the two waveforms appear basically coincide. Since the vibration displacement changes from negative to positive and passes through the equilibrium position, compared with the tip of the rigid rotating blade, the tip of the vibrating blade enters the sensing range of the sensor with a lag and exits the sensing range of the sensor first. Figure 4b, Case 2, shows the vibration ahead; the vibration speed changes from positive to negative and passes through the 0 position, and the vibration displacement is the largest, so the tip-timing waveform of vibrating blade is shifted to the  
left compared with the tip-timing waveform of rigid rotating blade--that is, the vibration is ahead. Compared with the tip of the rigid rotating blade, the tip of the vibrating blade enters and exits the sensing range of the sensor first. Figure 4c, Case 3, is the case of negative vibration synchronization. Vibration displacement changes from positive to negative through the equilibrium position, and the vibration displacement is close to 0, so the times when the highest points of the two waveforms appear basically coincide. Since the vibration displacement changes from positive to negative through the equilibrium position, compared with the tip of the rigid rotating blade, the tip of the vibrating blade enters the sensing range of the sensor first and exits the sensing range of the sensor with a lag. Figure 4d, Case 4, is the case of vibration hysteresis; the vibration velocity changes from negative to positive and passes through the 0 position, and the vibration displacement is the largest negative, so the tip-timing waveform of vibrating blade is shifted to the right compared with the tip-timing waveform of rigid rotating blade; that is, in vibration hysteresis, compared with the tip of the rigid rotating blade, the tip of the vibrating blade enters and exits the sensing range of the sensor with a lag. The influence of blade vibration amplitude on the blade tip-timing waveform is shown in Figure 5. Figure 5a, Case 1, shows the positive synchronization of vibration. When the blade vibration amplitudes are different, the times of highest points of tip-timing waveforms appear basically coincide. With the increase of vibration amplitude, the waveform shrinks inward and the time for the blade tip to pass through the sensing range of the sensor becomes shorter. Figure 5b, Case 2, shows the vibration ahead.