# Determining Rotor Blade Multi-Mode Vibration Components +
## 5 Numerical Analysis
The Appendix A Algorithm is again used here to accurately calculate the amplitudes and phases for these frequencies. Four cases of blade vibration were considered: first, when the blade vibrates only with the first frequency, 535 Hz; second, with two frequencies: 535 Hz and 1396 Hz; third, with three frequencies: 535 Hz, 1396 Hz and 1944 Hz; fourth, with four frequencies: 535 Hz, 1396 Hz, 1944 Hz and 3073 Hz. In the case of four blade vibration components k (Equation (1)), only the first one has a maximal amplitude of 0.27 mm, resulting from 2 EO. For a frequency of 1396 Hz, the blade amplitude is 0.05 mm, 1944 Hz (0.02 mm) and 3073 Hz (0.01). This shows that the rotor blades vibrate predominantly with one component. Table 3 shows that the higher the number of harmonics, the greater the accuracy of the calculations, though the improvement is only slight. Figures 11-14 present blade amplitudes A\({}_{\rm w}\) versus various frequencies \(\omega_{\rm w}\) from Equation (A15), see Appendix A, for rotation speed \(\Omega\) = 202.3 Hz = 12,138 rpm and a time step of \(\Delta\)t = 1 \(\omega_{\rm s}\). (Equation (A3)) and N = 50. In this case, the blade vibrations are higher than for \(\Omega\) = 250 Hz (see Figures 7-10) as a result of their non-synchronous motion. The results presented in Figures 11-14 allow us to determine the blade vibration frequencies \(\omega_{1}\) = 464, \(\omega_{2}\) = 1470 and \(\omega_{3}\) = 1902 Hz. Figure 12: Amplitude A\({}_{\rm w}\) versus \(\omega_{\rm w}\) = 500–1000 Hz for \(\Omega\) = 202.3 Hz. \begin{table}
\begin{tabular}{c c c c c} \hline
**i** & \(\omega_{\rm i}\) **(Hz)** & **A\({}_{\rm i}\) (m/s)** & \(\varphi_{\rm i}\) **(deg)** & **A\({}_{\rm ix}\) (mm)** \\ \hline
1 & 535 & 0.89 & 166 & 0.27 \\ \hline
1 & 535 & 0.9 & 166 & 0.27 \\ \hline
2 & 1396 & 0.45 & 164 & 0.06 \\ \hline
1 & 535 & 0.9 & 166 & 0.27 \\ \hline
2 & 1396 & 0.45 & 164 & 0.06 \\ \hline
3 & 1944 & 0.21 & 157 & 0.02 \\ \hline
1 & 535 & 0.9 & 166 & 0.27 \\ \hline
2 & 1396 & 0.4 & 170 & 0.05 \\ \hline
3 & 1944 & 0.2 & 157 & 0.02 \\ \hline
4 & 3073 & 0.26 & 56 & 0.01 \\ \hline \end{tabular}
\end{table} 
Table 3: The blade frequency \(\omega_{\rm i}\), velocity amplitude A\({}_{\rm i}\), phase \(\varphi_{\rm i}\), and amplitude A\({}_{\rm ix}\) for \(\Omega\) = 250 Hz. From the analysis, the following vibration frequencies were found: \(\omega_{1}\) = 464 Hz, \(\omega_{2}\) = 1470 Hz, and \(\omega_{3}\) = 1902 Hz when \(\omega_{\rm w}\) moves from 500 Hz to 2000 Hz, assuming one-mode blade vibration Equation (2). So there are three vibration components in Equation (1). These frequencies coincide with blade natural frequencies calculated using MES [16; 17]. As when \(\Omega\) = 250 Hz, here, too three cases were considered: first, when the blade vibrates only with the first frequency, 464 Hz; second, with two frequencies: 464 Hz and 1407 Hz; third, with three frequencies: 464 Hz, 1407 Hz and 1907 Hz (see Table 4). In the case of blades vibrating with three components, k = 3 (Equation (1)), the first, at 464 Hz, has a maximal amplitude of 0.81 mm. The amplitude is higher than for the nominal \begin{table}
\begin{tabular}{c c c c c} \hline
**i** & \(\omega_{1}\) **(Hz)** & **A\({}_{1}\) (m/s)** & \(\varphi_{1}\) **(deg)** & **A\({}_{\rm ix}\) (mm)** \\ \hline
1 & 464 & 2.36 & 101 & 0.81 \\ \hline
1 & 464 & 2.16 & 100 & 0.74 \\ \hline
2 & 1470 & 0.92 & 64 & 0.01 \\ \hline
1 & 464 & 2.15 & 99 & 0.74 \\ \hline
2 & 1470 & 0.91 & 62 & 0.10 \\ \hline
3 & 1902 & 1.10 & 71 & 0.09 \\ \hline \end{tabular}
\end{table} 
Table 4: The blade frequency \(\omega_{\
u}\) velocity amplitude A\({}_{\rm i}\), phase \(\varphi_{\rm i}\), and amplitude A\({}_{\rm ix}\) for \(\Omega\) = 202.3 Hz. condition (Table 3) as a result of rotation stall [15]. For 1470 Hz, it is 0.10 mm, and for 1902 Hz, it is 0.09 mm, again higher than for nominal conditions due to rotation stall. This shows that in the case of non-nominal conditions, the differences between blade amplitude components are smaller, which is evidence of multi-mode coupling.