## 3 Recognition Precision Verification  
### Effect of BTC Measurement Error on Recognition Precision  
The identification accuracy of the above method is verified by numerical simulation. It is assumed that a single-frequency synchronous vibration occurs when the blade speed is 3000 rpm. The blade excitation order \(N_{e}\) is 4, the amplitude \(A\) is 1-20 mm, the initial phase \(\varphi\) is 0 and the constant bias \(C\) is = 2 mm. Seven BTC sensors and a speed sensor are used to identify the blade vibration parameters. The installation angles of the sensors are \(0^{\circ},18.4^{\circ},36^{\circ},56^{\circ},72.3^{\circ},119.5^{\circ}\) and \(238.9^{\circ}\), respectively, as shown in Figure 2. When the blade length \(r\) is 50 mm, the radius of the blade disk \(e\) is 150 mm, and the distance between the tip of the BTC sensor and the rotation axis \(R\) is 202 mm. The measured values of the BTC sensor introduce \(\pm 0.1\) mm, \(\pm 0.01\) mm, \(\pm 0.001\) mm and \(\pm 0.0001\) mm, respectively, and have a uniformly distributed random error. The results of excitation order identification under different random errors and amplitude conditions are shown in Figure 3. When the random error of the BTC measurement is less than \(\pm 0.01\) mm, the excitation order is accurately identified as 4 using the above method. When the amplitude is greater than 6 mm and the random error of the BTC measurement is less than \(\pm\)0.1 mm, the excitation order is accurately identified as 4 using the above method. As the random error of BTC measurement decreases, the accuracy of vibration parameter identification of this method increases. The measured value of the BTC with an amplitude \(A\) of 1 mm is shown in Table 1, and the excitation order recognition result is shown in Figure 4. The measured values of BTC with the amplitude \(A\) of 6 mm are shown in Table 2, and the results of excitation order recognition are shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Random Error** & **Tip0** & **Tip1** & **Tip2** & **Tip3** & **Tip4** & **Tip5** & **Tip6** \\ \hline \(\pm 10^{-1}\) mm & 2+ & 2.1+ & 2+ & 1.9+ & 1.9+ & 2+ & 2.1+ \\ \(\pm 10^{-2}\) mm & 2.02+ & 2.05+ & 2.04+ & 2.02+ & 2.00+ & 2.06+ & 2.00 \\ \(\pm 10^{-3}\) mm & 2.027+ & 2.062+ & 2.047+ & 2.012+ & 2.005+ & 2.058+ & 2.006 \\ \(\pm 10^{-4}\) mm & 2.0263+ & 2.0620+ & 2.0466+ & 2.0119+ & 2.0047+ & 2.0589+ & 2.0069 \\ \hline Theoretical value & 2.026431+ & 2.062105+ & 2.046652+ & 2.011877+ & 2.004789+ & 2.058760+ & 2.006784+ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured BTC values when \(A\) is 1 (mm). (“+” for early arrival). Figure 3: Result of excitation order recognition when \(A\) is 1–20 mm. Figure 2: Sensor Installation Diagram. When the blade length \(r\) is 100 mm, the blade disk radius \(e\) is 100 mm, the distance between the tip of the BTC sensor and the rotation axis \(R\) is 202 mm. The measured values of the BTC sensor introduce \(\pm\)0.1 mm, \(\pm\)0.01 mm, \(\pm\)0.001 mm and \(\pm\)0.0001 mm, respectively, and have a uniformly distributed random error. The results of excitation order identification under different random errors and amplitude conditions are shown in Figure 6. When the random error of the BTC measurement is less than \(\pm\)0.001 mm, the excitation order is accurately identified as 4 using the above method. When the amplitude is greater than 4 mm and the random error of BTC measurement is less than \(\pm\)0.01 mm, the excitation order is accurately identified as 4 using the above method. When the amplitude is greater than 9 mm and the random error of BTC measurement is less than \(\pm\)0.1 mm, the excitation order is accurately identified as 4 using the above method. As the random error of BTC measurement decreases, the accuracy of vibration parameter identification by this method increases. Figure 4: Result of excitation order recognition when \(A\) is 1 mm. Figure 5: Result of excitation order recognition when \(A\) is 6 mm.