## 9 Multimode Analysis for Non-Nominal Conditions  
The L-M algorithm was used for a multimode analysis (Equation (3)) of rotor blades in measurements 1, 2 and 3. This showed that all the blades vibrated in the 1 st and 2 nd mode simultaneously, which is a novelty. Figures 21 and 22 present 1 st and 2 nd mode blade amplitudes and frequencies for measurement 1. The blade frequency results were similar for all three measurements. The figures show that the 2 nd mode of vibration frequency appeared at about 3.5 [-], which corresponds to the second blade natural frequency. The 3 rd mode did not appear. The 2 nd blade mode amplitude was significantly lower than the 1 st blade mode amplitude. The maximum 1 st mode blade amplitude was about 0.349 mm and 0.17 mm for the 2 nd mode for condenser pressure 23.5 kPa in measurement 1 (Figure 22). The 1 st blade amplitude for the 1 st mode was about 0.31 mm and 0.11 mm for the 2 nd mode for condenser pressure 23.5 kPa in measurement 1 (Table 2). For measurement 2 and condenser pressure 19 kPa (Table 3), the 1 st blade amplitude of the 2 nd mode was lower than for measurement 1  
The above multimode analysis results were compared with the results in [34], which presented the method of finding one-mode synchronous and nonsynchronous blade vibrations from the tip-timing velocity time of arrival using the least squares technique. In [34], two sensors were required in the casing without a once-per-revolution sensor and without measuring the rotor blade velocity. The rotor blade velocity was obtained from the measured time of rotor blade arrival and used in the functional of the least squares technique. In this paper, the above model was altered for a multimode analysis, which is a novelty. Tables 2-4 present the amplitudes and frequencies (measurements 1, 2 and 3, respectively) for the L-M algorithm (method 1) and the multimode analysis based on [34] (method 2). For measurement 1 (Table 2), the 1 st mode frequency was nearly identical (<1%), although the frequency obtained by method 2 was slightly higher. The amplitude was higher for the 1 st blade mode than for the 2 nd blade mode, and here there was also a greater disparity between methods 2 and 1. Similar conclusions may be drawn for the 1 st amplitude mode in measurements 2 (Table 3) and 3 (Table 4). For the 2 nd mode, the frequencies and amplitudes in method 1 were slightly higher. The general conclusion is that in method 1, the results were similar to those of method 2. The influence of the number of sensors on blade amplitude and frequency calculation accuracy was analyzed in the previous section. As already demonstrated, the L-M algorithm can analyze multimode blade vibrations. Such calculations increase the accuracy of amplitude calculations. Figures 21 and 22 show that the blades vibrated simultaneously with two modes in non-nominal regimes. A comparison between a one-mode blade vibration algorithm [27] and the L-M multimode algorithm shows that the blade vibration amplitude for the one-mode algorithm was higher (Figure 23). The one-mode analysis is an approximation of displacements comprising a few modes, which is not precise. Figure 22 presents the blade amplitude for 53 blades (measurement 1) with 10.67 kPa condenser pressure. The one-mode (SDOF) [27] blade amplitudes are red and the multimode (MDOF) 1 st mode amplitudes are green. The horizontal red line is the averaged SDOF blade amplitude [27] and the horizontal green line is the averaged multimode (MDOF) 1 st mode blade amplitude. This clearly shows that the one-mode averaged blade amplitude was 29% higher than that of the multimode 1 st mode. This is an estimation of the accuracy of the L-M algorithm for multimode blade vibrations. Its accuracy in calculating blade vibration amplitudes could be very important from the point of view of blade life. The procedure of life calculation is presented in [35].