## 4 Results of the Acoustic method for Mistuning Identification  
### Computational Finite Element Model of the Turbine Impeller  
The computational finite element model of one impeller segment used for its modal analysis is shown in Figure 2 left. For this sub-model, the boundary conditions were specified as zero displacements and zero rotations in all three coordinate axes for nodes at both edges. Thus, the eigenfrequencies and their mode shapes associated with the impeller blade were obtained. Some selected examples are shown in Figure 7.  
Figure 5: Impeller blade response to modal hammer force impulse—laser vibrometer measurement.  
Figure 6: Impeller blade response to modal hammer force impulse–microphone measurement.  
The calculated turbine blade segment eigenfrequencies are shown in Table 1. Due to the value of the maximum impeller speed, the eigenfrequencies were investigated in the frequency range up to 20 kHz, which was also the frequency range of the measuring microphone used.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Mode (-)** & **Frequency (Hz)** \\ \hline
1 & 4718 \\
2 & 7737 \\
3 & 9110 \\
4 & 11,333 \\
5 & 14,107 \\
6 & 14,990 \\
7 & 18,511 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Calculated eigenfrequencies of the impeller blade segment.  
Figure 7: The first six blade mode shapes of the impeller segment.  
These eigenfrequencies determine approximately the ranges of values where the blade-dominated mode families will be located for the entire impeller.  
The created mesh of the finite element model according to Figure 2 was further used to build a tuned rotational cyclic model of the whole turbine impeller. Its result is a nodal diameter map of the tuned impeller. Given the main topic of this article, these results are not presented in more detail.  
Using the rotation of one segment, a 3D finite element model of the turbine wheel was subsequently created, the mesh of which is identical to the rotational cyclic model. Detailed calculations have verified that the results of the modal analysis of the 3D impeller model and the rotationally cyclic model are completely identical, the values of all calculated eigenfrequencies are the same to at least two decimal places  
This model will continue to be used for analysis with different material properties of the blades and for studying the mistuning effect and for research of dangerous stress concentrators due to different characteristics of the individual blades. Figure 8 shows examples of the blade-dominated mode shapes of Modes 1-6. Due to the rigid structure of the impeller disc examined, there are no disc dominated mode shapes in the frequency range to 20 kHz. Instead, the vibration behaviour is determined by blade dominated and mixed mode shapes.  
In the context of the blade vibrations of turbochargers, the analysis of the blade dominated mode shapes is of interest since unsteady pressure fluctuations in operation excite mainly just these. Examples of two different types of bladed wheel vibrations are shown in Figure 9. It is striking that the optical comparison of the displacements provides little information about the type of vibration to which the respective mode shape can be assigned. Information on strain energy distribution is crucial for the correct classification of the mode type. In the case of Mode 1 it is a blade dominated shape, while in the case of Mode 6 it is a mixed mode (see Figure 9).  
Figure 8: _Cont._