**1. Introduction**

The principle of operation of the exhaust gas turbocharger is to use the unused kinetic and thermal energy of the ejected engine exhaust. The mostly single-stage turbine is set in rotation by the exhaust gas flowing out and thus drives the compressor. This increases the fluid pressure of the outflowing air which is then pressed into the engine. The increased cylinder filling of the driven engine enables better thermal efficiency during combustion. Turbochargers are also an important part of various systems for reducing emissions from internal combustion engines [1–10]. Due to the increased boost pressure of the working medium, either more powerful engines with similar dimensions or similarly powerful engines with significantly smaller dimensions are possible.

In general, particularly high pressure ratios of the compressor blades are required to achieve the required performance increases. The high aerodynamic demands require a very detailed design of blade geometries. Ever thinner blades combined with the low mechanical damping due to the integral construction make the vibration-proof design of turbomachines more di fficult. Regardless of the widespread usage of turbomachines, documented damage cases indicate that structural dynamic issues have not been conclusively resolved. Imperfections that already occur during production have a significant influence on the dynamic behaviour of real components. The deviation from the original design is referred to as a "mistuned bladed disc e ffect" and has been discussed by many researchers in the field of internal combustion engines and aircraft turbine engines [11–14]. Researchers [11,12] have described a method that allows mistuning distributions to be determined indirectly from vibration measurements without asking about the actual cause of the mistuning.

The largely random mistuning reduces the life of the components that are permanently loaded by centrifugal forces, flow deflection, unsteady pressure fluctuations in the flow and temperature gradients. The purely numerical prediction of the resonance strength is almost impossible for the entire operating range due to the numerous operating points. The vibration safety of a new development is therefore detected in regard to the qualification tests. As a rule, these consist of driving through the operating area of the exhaust gas turbocharger on a combustion chamber or an engine test benches. The measured vibrations of the system are included in the fatigue strength analysis. Considerable safety factors are taken into account. If detailed statements on the real structural behaviour of a design were available at an early stage, the safety factors of some resonance points could be reduced. As a result, designs would be feasible that are closer to the permissible mechanical load limit. Only sound assessments guarantee a solid and resonance-proof turbocharger design.

Knowledge of the real deviation between adjacent blades is necessary for the robust consideration of random mistuning. The parameters can be determined for a single, actually existing rotor using a suitable experimental method. A variety of approaches have now been documented. Some authors [15,16] measured the surface of the manufactured blades using white light stripe projection. In doing so, they directly countered the geometric manufacturing tolerance as a presumed cause of the mistuning. Another publication [17] continued this development and demonstrated good agreemen<sup>t</sup> between the properties of optically recorded blades and experimental vibration measurements. Some research studies have also appeared in the field of aircraft and automotive turbochargers [18,19], but their results are not very applicable, particularly with regard to the point of identification of the mistuning and the cause of the mistuning of marine engine turbocharger bladed discs.

Further research to identify mistuning has been done mainly in the field of aircraft engines [20]. This reflects the current trend of applying the integrated structures of blade and disc. Contemporary technology enables this type of structure, which has the advantage of e ffectively reducing both the weight and the number of engine parts. Contrariwise, there are di fficulties with mistuning identification. The reason is that with such a structure, separation of each blade from the disc is not possible, as with the dovetail type blade of conventional compressors. Therefore, it is not possible to know the natural frequency of each blade without disc coupling e ffect. Some authors dealt with the mistuning problems of small rotors of automotive turbochargers [21,22]. One paper [23] presented the experimental and numerical studies of last-stage low-pressure mistuned steam turbine bladed discs during run-down. The tip-timing method was used to find the mistuned bladed disc modes and frequencies.

Based on this situation, a marine engine turbocharger rotor was selected as the subject of the research (see Figure 1). The aim of the solution was to identify mistuning of actually produced turbine rotor of a marine engine turbocharger and to design an e ffective experimental method for this purpose. Some authors used a laser scanning system to obtain the modal information for di fferent types of mechanical structures [24–27]. In this system, a vibration response against the excitation input is obtained at each reference point using a laser Doppler vibrometer. This process is repeated for the same excitation input, the reference point being changed by scanning the laser Doppler vibrometer. After completion of the scan, the frequency response functions obtained for all reference points are correlated and the modal displacement contour of the entire measured surface is created. This procedure requires the application of a very expensive laser scanning technique; moreover, the processing of the measured

results is time consuming. Therefore, a procedure was proposed that uses only a simple laser vibrometer and an FE computational model of the turbine rotor.

**Figure 1.** Marine engine turbocharger rotor.

### **2. Modelling of Tuned Blade Disc Systems**

Modelling using finite elements makes dynamic analysis of complex components considerably easier. The vibration behaviour of turbocharger blade discs can be described by the differential motion equation:

$$M\ddot{q} + R\dot{q} + Kq = F(t),\tag{1}$$

where *M* is mass matrix, *R*—damping matrix, *K*—stiffness matrix, *F*—vector of force, *q*—vector of generalised coordinates and *t*—time in the continuous structure being discretised by a finite number of degrees of freedom. The characteristic values of the natural vibration behaviour are the eigenvalues, i.e., natural frequencies and eigenvectors, i.e., mode shapes. These can be determined by solving the homogeneous and undamped system according to Equation (1). If the vibration behaviour of the bladed disc is to be described as precisely as possible, a very fine discretization may be required. When using FEM, models with several million degrees of freedom are quickly created. Such model sizes are seamlessly manageable currently, but the question of reducing the degrees of freedom of the model arises without losing accuracy in the system description. In addition to classic reduction methods, the exploitation of symmetry properties is essential. With an ideal bladed disc, all sectors are identical, i.e., they have the same mechanical properties, so that no distinction among the stiffness matrices of the individual sectors is required.

The cyclic symmetry of the blade disc basically allows the analysis to be restricted to one sector without loss of information. Figure 2 shows one sector of the overall blade disc structure. The element size represents a central setting parameter that has a direct influence on the calculation results. The number of elements increases with an increasing level of model details. This is associated with an increasing number of modes to be calculated, which in turn increase the computational demands. In the case of the radial turbine investigated here, a fine and structured mesh of the blade surface is recommended. The disc body and shaft can be modelled using larger elements. The FE model of the entire turbine blade disc is created by the periodic expansion of the sector and subsequent fusion of the sector boundaries. The FE mesh was created at ANSA software and it had to be done thoroughly in connection with the subsequent creation of a rotation model. There, it is important to properly connect the nodes on both sides of the segmen<sup>t</sup> to create the whole rotational cyclic model. At the same time, it is necessary to achieve just the same segments that are really connected correctly. The size of the elements used was carefully optimised by a series of sensitivity analysis.

**Figure 2.** An illustration of one sector and whole mesh of the turbine bladed disc structure.

The turbine rotor is supported in bearings during operation. In modern exhaust gas turbochargers, two plain bearings prevent radial displacements. In addition, a thrust bearing is fitted. The bearing influence plays a negligible role with regard to the blade-dominated vibration forms. In order to achieve the best possible correlation, no boundary conditions are specified for the computational modal analysis.
