# Analysis of the Mode Shapes of Kaplan Runners
## 6 Conclusions  
In this study, the mode shapes of Kaplan runners have been thoroughly analyzed and discussed. To understand the modal characteristics of these bladed structures, four different bladed disks, with 4, 5, 6 and 7 blades, have been studied as simplified models of a Kaplan runner. The mode shapes of bladed disks have been classified according to the number of nodal diameters identified, using the mode shapes of a simple disk as a reference. The mode shapes of a bladed disk model and a real runner geometry have been compared using numerical simulations. Next, the results obtained from the real Kaplan simulation have been validated using experimental data obtained from an experimental modal analysis. By the tracking and identification of the disk nodal diameters on the bladed disks, it is found that the blade disk mode shapes are dependent on the blade number. On one hand, bladed disks with an even blade number have a uniform deformation distribution in the mode shape when the mode has (k*Z)/2 nodal diameters (with k = 1, 2, 3... and Z is the blade number). On the other hand, bladed disks with an odd blade number have a uniform deformation distribution, when the mode has k*z nodal diameters. In other cases, the mode shapes match the structure by a non-uniform deformation of the blades. Then, bending and torsional modes of the blades can coexist in a global mode shape, vibrating at different amplitudes. A classification of the mode shapes of bladed disk has been carried out. The disk mode shapes have been extracted and its nodal diameters have been tracked in the progressive modification of the disk to model a bladed disk. It was found that not only the disk nodal diameters can be seen in the blades but also in the phase shift between the blades. Then, the mode shapes can be classified as disk mode shapes. The equivalent nodal diameters number is obtained by adding the number of nodal diameters which pass through the blades and the diameters which provokes the counter phase motion between the blades. Using numerical simulation, it has been proved that the bladed disk model with 4 blades and a real Kaplan runner with the same blade number have similar mode shapes. This was validated by comparing the numerical results with the obtained mode shapes from experimental modal analysis. In the experimental results, global and local mode shapes have been identified and classified. Global mode shapes experimentally found are dominated by the first bending (1B) and first torsion (1T) motion of the blades. In addition, local mode shapes experimentally found, have been investigated by numerical methods. It is found that slight stiffness variations in the blades can affect the global mode shapes of the runner leading to the generation of local mode shapes. In this paper, a complete description of mode shapes of Kaplan runners and simplified models is given. This information is useful for the reader to understand the influence of parameters, such as the blade number, on the global mode shapes in the design stage of the \begin{table}
\begin{tabular}{c c c c} \hline \hline
34.593 Hz & 37.977 Hz & 38.391 Hz & 43.913 Hz & 49.393 Hz \\ \hline
50.498 Hz & 50.554 Hz & 52.438 Hz & 52.712 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 20: Mode shapes extracted after a blade stiffness variation. runner. Moreover, with this information, an estimation of the mode shapes which can be excited, e.g. if the runner is excited by a rotor-stator interaction, can be carried out. Writing-original draft preparation, G.M. ; Investigation and numerical study of the bladed disk, G.M. ; Numerical simulation of the Kaplan rotor, M.E. ; Experimental Modal Analysis, A.P. and D.V. ; Supervision, C.V. All authors have read and agreed to the published version of the manuscript. This work was funded by the National Agency for Research and Development (ANID)/Scholarship Program/DOCTORADO BECAS CHILE/ 2020-72210054. Furthermore, this research activity is framed within the context of the XFLEX HYDRO project.