# Analysis of the Mode Shapes of Kaplan Runners
## 3 Natural Frequencies and Mode Shapes of Simplified Structures
### Influence of the Blade Number on the Mode Shapes
#### 3.3.2 Diametrical Mode Shapes of Bladed Disks  
According to the literature [27], after a numerical modal analysis is performed, a pair of diametrical mode shapes is extracted at the same natural frequency. These modes have the same mode shape but different angular orientations, as shown in the \"Disk\" column in Tables 5-10. But then, when the disk is progressively modified, the equal diametrical mode pair of the disk become two different global modes in the bladed disk (see the aforementioned tables). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Disk** & **Z = 4** & **Z = 5** & **Z = 6** & **Z = 7** \\ \hline \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Disk** & **Transition 1** & **Transition 2** & **Transition 3** & **Bladed Disk** \\ \hline
600.80 Hz & 583.52 Hz & 520.45 Hz & 313.16 & 162.84 Hz \\ \hline
600.80 Hz & 583.99 Hz & 539.40 Hz & 430.09 Hz & 224.71 Hz \\ \hline \end{tabular}
\end{table} 
Table 7: Mode (4,0) for a bladed disk with \(Z=4\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Disk** & **Transition 1** & **Transition 2** & **Transition 3** & **Bladed Disk** \\ \hline
1050.00 Hz & 985.55 Hz & 778.21 Hz & 335.10 Hz & 160.50 Hz \\ \hline
1050.00 Hz & 995.04 Hz & 896.11 Hz & 747.68 Hz & 652.96 Hz \\ \hline \end{tabular}
\end{table} 
Table 5: Mode (2,0) for a bladed disk with \(Z=4\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Disk** & **Transition 1** & **Transition 2** & **Transition 3** & **Bladed Disk** \\ \hline
3010.90 Hz & 2797.8 Hz & 2107.70 Hz & 1060.60 Hz & 319.08 Hz \\ \hline
3010.90 Hz & 2840.2 Hz & 2545.90 Hz & 2207.40 Hz & 1748.70 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Mode (7,0) for a bladed disk with Z = 7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Disk** & **Transition 1** & **Transition 2** & **Transition 3** & **Bladed Disk** \\ \hline
2259.10 Hz & 2099.6 Hz & 1572.40 Hz & 772.65 Hz & 254.37 Hz \\ \hline
2259.10 Hz & 2131.30 Hz & 1913.30 Hz & 1694.40 Hz & 1447.6 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Mode (6,0) for a bladed disk with Z = 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Disk** & **Transition 1** & **Transition 2** & **Transition 3** & **Bladed Disk** \\ \hline
1604.30 Hz & 1532.00 Hz & 1261.80 Hz & 524.68 Hz & 200.53 Hz \\ \hline
1604.30 & 1540.70 Hz & 1421.50 Hz & 1315.60 Hz & 1185.30 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Mode (5,0) for a bladed disk with Z = 5.  
possible to see that the nodal diameters are placed in two main directions: Between the blades (Table 5, row 1) and across the blades (Table 5, row 2). When the nodal diameter passes through the blade, the vibration of each blade is associated with a torsional movement. In contrast, when the nodal diameter is placed in between the blades, the vibration of each blade is associated with a bending movement. The same effect occurs when a bladed disk with Z = 6 vibrates at the mode (3,0) (see Table 6). In here, the torsional and bending vibrations in the blades are also determined by the position of the nodal diameters with respect to them. Then, when the number of nodal diameters is half the number of blades (m = Z/2), the global mode shapes are associated with a bending mode shape and a torsional mode shape of the blades. Furthermore, as it can be seen in the \"Bladed Disk\" column in Tables 5 and 6, each blade moves in counter-phase with respect to the next blade. Now, when the bladed disk with Z = 4 vibrates in the mode (4,0), as shown in Table 7, the geometry of the structure and the number of nodal diameters distribution are also compatible. It can be seen that mode shapes of the bladed disk are also dependent on the orientation of the nodal diameters. In this case, also two orientations are possible. In the first one, the nodal diameters pass through the blades and in between them, which is related to a torsional vibration of each blade as a rigid body. In the second orientation, the nodal diameters pass only through the blades.