# Determining Rotor Blade Multi-Mode Vibration Components +
## 3 Validation of the Algorithm by Numerical Simulation  
The purpose of this simulation is to verify the algorithm presented in this paper. It was assumed that the rotor speed was \(\Omega\) = 15,000, rpm = 250 Hz, and the rotor blade vibrated with the assumed parameters in Table 1, i.e., with 4 harmonics, the velocity amplitudes \(\mathrm{A_{ie}}\), frequencies \(\omega_{\mathrm{ie}}\), and phase shift angles \(\varphi_{\mathrm{ie}}\). Having assumed the blade vibration parameters (Table 1), the blade time of arrival was calculated by two sensors mounted at 20 deg. angles in the compressor casing and a blade tip radius R = 0.314 m.  
First to be determined were the blade velocity, based on times of arrival \(\mathrm{t_{1n}}\), \(\mathrm{t_{2n}}\),... then \(\mathrm{t_{v1,n}}\), \(\mathrm{t_{v2,n}}\),... (Equation (A11)) and velocities \(\mathrm{v_{1,n}}\), \(\mathrm{v_{2,n}}\),... (Equation (A11)). The above data were determined with the time step \(\mathrm{\Delta t}\) = 1 \(\mathrm{\mu s}\) (Equation (A3)). These were used to determine the blade vibration parameters a (Equation (A10)), velocity amplitudes \(\mathrm{A_{i}}\), displacement amplitudes \(\mathrm{A_{xi}}\), phase angles \(\varphi_{\mathrm{i}}\) (Equation (A7)), velocity V (Equation (A9)), and displacement X (Equation (A14)). The velocity amplitudes \(\mathrm{A_{i}}\), frequencies \(\omega_{\mathrm{i}}\), and phase shift angles \(\varphi_{\mathrm{i}}\) are compared with \(\mathrm{A_{ie}}\), \(\omega_{\mathrm{ie}}\), \(\varphi_{\mathrm{ie}}\) assumed in Table 1 to validate the algorithm. Next, we can calculate the accuracy of the vibration velocity amplitude:  
\[\varepsilon_{\mathrm{iA}}=(\mathrm{A_{ie}}-\mathrm{A_{i}})/\mathrm{A_{ie}} \tag{3}\]  
where \(\mathrm{A_{ie}}\) is the blade amplitude from the simulation (Table 1) and \(\mathrm{A_{i}}\) is the calculated blade amplitude using the algorithm in the Appendix A. Phase accuracy:  
\[\varepsilon_{\mathrm{i\varphi}}=(\varphi_{\mathrm{ie}}-\varphi_{\mathrm{i}})/ \varphi_{\mathrm{ie}} \tag{4}\]  
where \(\varphi_{\mathrm{ie}}\) is the phase from the simulation (Table 1) and \(\varphi_{\mathrm{i}}\) is the calculated phase using the algorithm in the Appendix A. As presented in Equation (1) and in the Appendix A, the developed method of identifying the blade vibration velocity amplitudes \(\mathrm{A_{i}}\), blade displacements \(\mathrm{A_{xi}}\) and phase angles \(\varphi_{\mathrm{i}}\) requires the assumption of vibration frequencies that correspond to the number of harmonics. In the simulation, it was assumed that the blade vibrated with 4 harmonics for the parameters given in Table 1.  
We used the algorithm (Appendix A) to calculate the number of harmonics and compared the results with those assumed in Table 1.  
In Figures 1-4, the blade velocity amplitude \(\mathrm{A_{w}}\) [m/s] versus frequency \(\omega_{\mathrm{w}}\) [Hz] are presented with \(\omega_{\mathrm{w}}\), varying from 500 Hz to 2500 Hz. These figures show that the local extremes of \(\mathrm{A_{wi}}\) occur for frequencies \(\omega_{\mathrm{wi}}\). In Figure 1, the following frequencies [Hz] are seen: 766-\(\Omega\), 9\(\Omega\)-1720, 560, 12\(\Omega\)-2460,..., 766, 10\(\Omega\)-1720. The 560 Hz frequency is the \begin{table}
\begin{tabular}{c c c c} \hline
**i** & \(\mathrm{A_{ie}}\) **(m/s)** & \(\mathrm{\omega_{ie}}\) **(Hz)** & \(\varphi_{\mathrm{ie}}\) **(deg)** \\ \hline
1 & 1.5 & 560 & 10 \\ \hline
2 & 1.3 & 766 & 50 \\ \hline
3 & 0.6 & 1720 & 30 \\ \hline
4 & 0.5 & 2410 & 20 \\ \hline \end{tabular}
\end{table} 
Table 1: Assumed blade velocity amplitudes \(\mathrm{A_{ie}}\), frequencies \(\omega_{\mathrm{ie}}\) and phase \(\varphi_{\mathrm{ie}}\) for 4 harmonics. The calculated amplitudes are close to the assumed amplitudes \(\rm A_{ie}\) (Table 1). Next, the algorithm was used for four harmonic frequencies 560, 766, 1720 and 2410 Hz to find the blade velocity amplitudes \(\rm A_{ei}\), blade displacements \(\rm A_{xi}\) and phase angles \(\rm\varphi_{i}\) (Table 2).