# Determining Rotor Blade Multi-Mode Vibration Components +
## 5 Numerical Analysis  
This chapter presents a numerical analysis of blade vibrations in the 1 st-stage compressor of a one-pass SO-3 ISKRA engine. In several cases, blade failure caused engine damage and plane crashes. In order to obtain a full understanding of the measurement results, it was necessary to carry out numerical analyses using the Finite Element Method (FEM). The blade was modelled using Solid 45, isoparametric elements with eight nodes [14,15]. The mesh was as follows: one element for the thickness, 13 elements for the blade chord, and 10 levels for the blade length. This mesh was sufficient to obtain convergence with the experimental measurements [15]. The blade was made of 18H2N2 (structural chromium-nickel alloy steel for carburizing). In the calculations, it was fixed in the blade root. Its natural frequencies were compared with experimental ones using strain-gauges and tip-timing analysis [15]. The free vibration of the rotor blade was calculated using an FEM [14,15]. The first four natural frequencies at 15,000 rpm are 509.17, 1527.7, 1863.1, and 3127.8 Hz. From [15], it was known that the forced and non-synchronous (caused by rotating stall) blade vibration frequencies would be close to their natural frequencies. The rotor blade frequencies in the 1 st-stage compressor varied due to manufacture tolerances, e.g., 535, 512, 525, 524, 525, 525, 516, 525, 524, 515, 527, 518, 519, 544, 522, 517, 529, 518, 509, 523, 529, 509, 517, 518, 524, 507, 524, and 525 Hz. These frequencies correspond to the first bending blade frequency. For \(\Omega\) = 12,128 rpm, 464, 399, 408, 406, 405, 402, 392, 408, 404, 391, 396, 396, 393, 413, 400, 390, 395, 395, 395, 407, 402, 398, 406, 401, 397, and 395 Hz. Here, only rotor blade 1 was considered, as an example, but calculations for other blades are analogous. The blade vibration amplitude A\({}_{\mathrm{w}}\) and phase \(\varphi_{\mathrm{w}}\) can be found based on Equation (1) for rotation frequencies 250 Hz (15,000 rpm--nominal speed) and \(\Omega\) = 202.3 Hz (12,138 rpm--non-nominal speed), time step \(\Delta t\) = 1 \(\upmu\)s. (Equation (A3)) and [11] and N = 50. Figures 7-10 present blade amplitudes A\({}_{\mathrm{w}}\) versus various frequencies \(\omega_{\mathrm{w}}\) (Equation (2)) for rotation speed \(\Omega\) = 250 Hz. Axis x shows \(\omega_{\mathrm{w}}\) and axis y shows A\({}_{\mathrm{w}}\). Figure 6: Vibration of the 28 1 st-stage rotor blades rotating from 7500 to 16,000 rpm. Figure 8: Amplitude \(A_{\rm w}\) versus \(\omega_{\rm w}\) = 1000-2000 Hz for \(\Omega\) = 250 Hz. Figure 7 shows the amplitude A\({}_{\rm w}\) for \(\omega_{\rm w}\) from 100 to 1000 Hz. As Figures 1-4, Section 3, we found that \(\omega_{1}\) = 535 Hz (red) is the first frequency component in Equation (2) close to the natural frequencies. Figure 8, where \(\omega_{\rm w}\) is from 1000 Hz to 2000 Hz, shows that \(\omega_{2}\) = 1396 Hz and \(\omega_{3}\) = 1944 Hz. In Figure 9, \(\omega_{\rm w}\) frequencies change from 2000 Hz to 3000 Hz, but blade vibration components are not found. Figure 10, where \(\omega_{\rm w}\) changes from (3000 Hz-4000 Hz), \(\omega_{4}\) = 3073 Hz. Therefore, the spectrum of blade frequencies obtained from Figures 7-10 are \(\omega_{1}\) = 535 Hz, \(\omega_{2}\) = 1396 Hz, \(\omega_{3}\) = 1944 Hz and \(\omega_{4}\) = 3073 Hz. Similarly to Figures 1-4, additional frequencies \(\omega_{\rm w1}\) = \(\omega_{1}\) \(\pm\) j\(\Omega\), \(\omega_{\rm w}\) = j\(\Omega\) \(-\)\(\omega_{1}\) (j = 1,2, \(\ldots\), i = 1,2, \(1\) = \(\ldots\)) also appeared. From the analysis, the following vibration frequencies were found: \(\omega_{1}\) = 535 Hz, \(\omega_{2}\) = 1396 Hz, \(\omega_{3}\) = 1944 Hz, \(\omega_{4}\) = 3073 Hz for \(\omega_{\rm w}\) up to 4000 Hz, assuming one-mode blade vibrations (Equation (2)). So there are four vibration components in Equation (1). These frequencies coincide with the natural blade frequencies calculated using MES [14,15].