## 5 Verification of Tip-Timing Code  
In the first step, the L-M tip-timing code is verified using a rotor blade simulator [31]. In the next step, the algorithm is verified for nonsynchronous mistuned blade vibrations in a non-nominal regime (Table 1, measurement 1) for condenser pressure at 23.5 kPa (Figure 4) and 10.67 kPa (Figure 5). In mistuned blades, in this case, differences between amplitudes were small. LP last stage mistuned blade discovery nonsynchronous blade vibrations were measured by [PERSON] et al. [27]. The obtained rotor blade amplitudes were compared with the nonsynchronous tip timing results in [27] for one-mode blade vibrations. For this purpose, one mode of blade vibration is assumed in the L-M algorithm (Equation (1)). The comparison was satisfactory, with below 10% differences for most blades, Figure 4. The blue bars are the one-mode amplitudes from the L-M algorithm and the red ones are the results from [27]. In the case of lower condenser pressure (10.67 kPa, measurement 1), the differences between blade amplitudes were below 10%, Figure 5. The blue bars are the amplitudes from the L-M algorithm for one mode and the red ones are the results from [27]. The synchronous blade vibrations of an LP last stage mistuned blade discovery were measured in a vacuum spin chamber during run-down ([PERSON] et al. [6; 26], [PERSON] [32]). Single-coil inductive sensors detected blade tip deflections, while every revolution was signaled by a once-per-revolution sensor located between the rotor front support and stage. Fourteen sensors were installed above rotor blades in seven positions (seven pairs of sensors at different axes) [32]. The blades were excited using a manually regulated airflow from a telescopic pipe with a suitably shaped nozzle. The test was carried out during run-down (synchronous vibration). The tip displacements of 53 blades during a reduction of speed from 2028 rpm to 1516 rpm [6; 26] were analyzed. Figure 6 presents the blade amplitudes obtained from the L-M algorithm (blue) and from [26] (red). Figure 7 presents the frequencies. The comparison is satisfactory, with differences below 10% for the majority of the analyzed turbine blades. Figure 4: Comparison of Levenberg–Marquardt algorithm and one mode algorithm [27] results for 53 blade amplitudes in 23.5 kPa condenser pressure (Measurement 1). Figure 5: Comparison of Levenberg–Marquardt algorithm and one mode algorithm [27] results for 53 blade amplitudes in 10.67 kPa condenser pressure (Measurement 1). Figure 6: L–M algorithm and one mode algorithm [26] results for the amplitudes of 53 blades for EO3, 2028 rpm to 1516 rpm. The Influence of the Number of Sensors on the Accuracy of the Nonlinear Least Squares Levenberg-Marquardt Tip-Timing Fitting Method  
One of the main advantages of the nonlinear least-squares fitting method for tip-timing displacements is the fact that it requires only two sensors to give accurate results. Figure 8 shows that the differences between two sensors and three sensors for synchronous vibrations were below 1%. in normalized frequencies. Rotor blade normalized frequencies for 3, 4 and 5 sensors varied no more than 0.4% (Figure 9). For rotor blades amplitudes, the differences between two sensors and three sensors were around 10% (Figure 10). When the number of sensors was increased to four and then to five (Figure 11), the differences between amplitudes for three, four and five sensors were around 1-2%. Figure 7: L–M algorithm and one mode algorithm [26] results for the frequencies of 53 blades, 2028 to 1516 rpm and EO3. Figure 8: L-M algorithm normalized frequencies with 2 and 3 sensors, for 53 blades, 2028–1516 rpm and EO3. Figure 9: L-M algorithm normalized frequencies for 3, 4 and 5 sensors, for 53 blades, 2028 to 1516 rpm and EO3. The influence of the number of sensors on the accuracy of obtaining blade amplitude and frequency have been analyzed. However, the accuracy of the L-M method in comparison to one-mode analysis will be presented in the last section.