*4.1. Finite Elements Method*

Modal analysis of one turbocharger turbine segmen<sup>t</sup> was performed using FEM, as shown in Figure 7. Boundary conditions with zero displacement and rotation in all directions were applied to the sides of the segment. Thus, the natural frequencies associated with the turbocharger blade were obtained. In this way, the mode shapes were obtained. Figure 7 shows mode 2 for illustration.

**Figure 7.** Example of a mode shape of a turbocharger turbine blade segmen<sup>t</sup> (mode 2).

The calculated natural frequencies of the blade are presented in Table 1. The interest in the frequency range was up to 20,000 Hz, where the first seven modes are located. The calculated natural frequencies show approximate natural frequency values of the blade-dominant mode shapes of the rotational symmetric turbine disc.


**Table 1.** Calculated natural frequencies of the blade segment.

Subsequent forms of mesh models were used in the next step (Figure 2). The second analysis was performed from one segmen<sup>t</sup> as a rotational cyclic modal analysis. It results in a nodal diameter map of the tuned turbine rotor. However, this is not the main subject of this article.

Subsequently, a third FE model was created from a rotated model of one segment. The modal properties of this three-dimensional (3D) turbine FE model are identical to the previous rotational cyclic model. This model will further serve for analyses with different material properties of the blades and for analysing the mistuning effect. An example of the blade-dominated mode shapes for modes 1–6 is shown in Figure 8.

**Figure 8.** Examples of computed modal shapes of the turbocharger turbine wheel.

The material properties have been the same thus far for all segments to verify the following measurement results and possibly debug the model. The mistuned 3D model will be used for research of dangerous stress concentrators due to different characteristics of the individual blades.
