# Analysis of the Mode Shapes of Kaplan Runners
## 4 Modal Characteristics of a Kaplan Runner  
To complete the above analysis, the mode shapes of bladed disks are compared with the mode shapes of a real Kaplan runner with 4 blades (see Figure 4a). To make this comparison, a modal analysis of a Kaplan rotor is carried out by means of numerical simulation. The geometry of the rotor includes the runner, with its inner control system, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **(2,0)1B** & **(2,0)1T** & **(4,0)1T** & **(4,0)2T** \\ \hline ND & 0 & 2 & 2 & 0 \\ \hline cPD & 2 & 0 & 2 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Z = 4: Mode shapes classification. Figure 3: Torsional modes: (**a**) Blades without a nodal diameter between them, (**b**) blades with a nodal diameter between them. (see Figure 4b), the shaft and the generator. An Elastic Support condition with a Support Stiffness of 109 N/mm\({}^{3}\) is used. This boundary condition is placed in the thrust bearing location (see Figure 4c). In Table 15, the main dimensions of the rotor, indicated in Figure 4c, are defined as a function of the runner outer diameter Do. For the mesh, 931562 tetrahedral elements are used. The elements have a size of 70 mm in the blades, 30-70 mm in the inner control system, 100 and 120 mm in the hub and the cone, respectively, and 200 mm in the shaft and generator. These element size values were selected after an analysis of the mesh sensitivity was done. Next, the mode shapes and natural frequencies of the rotor are extracted. Even though the whole rotor is analyzed, its mode shapes are locally defined by its components (runner, shaft and generator). In Table 16, different mode shapes of the rotor are shown. These mode shapes are classified depending on the dominance of the rotor component on the mode shape. In the first row of the table a mode shape dominated by the deflection of the shaft is shown. In this mode shape, the runner moves as a rigid body. In the second row, the mode shapes dominated by the generator are shown. It can be seen that, due to its geometry, its mode shapes are related to disk-like structure mode shapes. Although these are generator-dominated mode shapes, it can be seen that the blades are also excited in global and local mode shapes. Finally, in the third and fourth rows, the ten first mode shapes dominated by the runner are shown. These last mode shapes are the ones studied in this paper. Figure 4: Kaplan runner model: (**a**) Runner, (**b**) Runner section plane, (**c**) Rotor main dimensions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dimension** & **Symbol** & **Percentage of \(D_{o}\) [\%]** \\ \hline Runner Outer Diameter & \(D_{o}\) & 100 \\ Runner Inner Diameter & \(D_{i}\) & 42 \\ Shaft Outer Diameter & \(D_{so}\) & 12.38 \\ Shaft Inner Diameter & \(D_{si}\) & 6.99 \\ Shaft Length & \(L_{s}\) & 96.18 \\ Generator Diameter & \(D_{g}\) & 131.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Kaplan rotor main dimensions. _Runner Mode Shapes and their Classifications_  
For a better analysis of the runner mode shapes, the shaft and generator of the rotor are not represented. In Table 17, a comparison of mode shapes of the bladed disk and the Kaplan runner is shown. In a general view, the modal characteristics of the bladed disk and the runner are similar. Although three mode shapes of the runner were not found in the bladed disk, most of the mode shapes of the runner match the mode shapes of the bladed disk. Then, the mode shapes of the Kaplan runner can be classified with the same criteria proposed for bladed disks. \begin{table}
\begin{tabular}{c c c c} \hline \hline Shaft & & & \\ Dominated & & & \\ Mode shape & & & \\ \hline \hline Generator & & & \\ Dominated & & & \\ Mode Shapes & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 16: Mode shapes of the Kaplan rotor. The unidentified mode shapes are classified as (1,0)1B, (3,0)1T1B and (4,0)1T-1 (see Table 17). In the first unidentified mode shape, 1 nodal diameters is detected. While in the second mode shape 3 nodal diameters are detected.