# Analysis of the Mode Shapes of Kaplan Runners

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical and Experimental Modal Analysis

- ω
_{r}is the natural frequency of the corresponding vibration mode. - θ
_{r}is the damping factor of the structure. - {θ}
_{r}is the vibration mode shape that dominates the structure near a resonance condition (ω_{excitation}= ω_{r}). - Q
_{r}is the scaling factor, which is a constant value for each mode shape.

- Receptance function: relation between the displacement and the excitation force.
- Mobility function: relation between the velocity and the excitation force.
- Inertance function: relation between the acceleration and the excitation force.

## 3. Natural Frequencies and Mode Shapes of Simplified Structures

#### 3.1. Modal Characteristics of a Disk-like Structure

#### 3.2. Transition from a Disk to a Bladed Structure

#### 3.3. Influence of the Blade Number on the Mode Shapes

#### 3.3.1. Mode Shapes Due to the Clamping

#### 3.3.2. Diametrical Mode Shapes of Bladed Disks

#### 3.3.3. Circular Mode Shapes of Bladed Disks

#### 3.4. Mode Shapes Classification

## 4. Modal Characteristics of a Kaplan Runner

^{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.

#### Runner Mode Shapes and their Classifications

## 5. Experimental Modal Analysis of a Kaplan Runner

#### 5.1. Instrumentation

#### 5.2. Runner Mode Shapes

#### 5.3. Local Mode Shapes

#### 5.4. Variation of Stiffness

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Detailed Geometries of Bladed Disk and Its Transitions

Z | Transition 1 | Transition 2 | Transition 3 | Final Model |
---|---|---|---|---|

4 | ||||

5 | ||||

6 | ||||

7 |

## Appendix B. Mode (1,0), (2,0) and (3,0) for All Transition Geometries and Bladed Disks

Z | Transition 1 | Transition 2 | Transition 3 | Final Model |
---|---|---|---|---|

4 | ||||

167.55 | 166.92 | 150.75 | 104.52 | |

5 | ||||

167.47 | 167.36 | 158.63 | 118.19 | |

6 | ||||

167.57 | 167.43 | 162.73 | 133.65 | |

7 | ||||

167.62 | 167.66 | 164.79 | 145.46 |

Z | Transition 1 | Transition 2 | Transition 3 | Final Model |
---|---|---|---|---|

4 | ||||

264.98 Hz | 245.03 Hz | 152.50 Hz | 101.26 Hz | |

266.55 Hz | 254.25 Hz | 216.49 Hz | 144.87 Hz | |

5 | ||||

268.07 Hz | 257.62 Hz | 188.58 Hz | 120.32 Hz | |

268.07 Hz | 257.62 Hz | 188.59 Hz | 120.31 Hz | |

6 | ||||

267.42 Hz | 254.97 Hz | 208.35 Hz | 143.41 Hz | |

267.42 Hz | 254.97 Hz | 208.35 Hz | 143.41 Hz | |

7 | ||||

268.08 Hz | 257.89 Hz | 220.39 Hz | 163.58 Hz | |

268.08 Hz | 257.90 Hz | 220.38 Hz | 163.58 Hz |

Z | Transition 1 | Transition 2 | Transition 3 | Final Model |
---|---|---|---|---|

4 | ||||

576.89 Hz | 507.55 Hz | 283.68 | 154.12 Hz | |

577.27 Hz | 510.25 Hz | 283.67 Hz | 154.13 Hz | |

5 | ||||

586.92 Hz | 541.60 Hz | 336.81 Hz | 177.91 Hz | |

586.93 Hz | 541.60 Hz | 336.80 Hz | 177.91 Hz | |

6 | ||||

583.52 Hz | 520.45 Hz | 313.76 Hz | 163.84 Hz | |

583.99 Hz | 539.40 Hz | 430.09 Hz | 224.71 Hz | |

7 | ||||

586.53 Hz | 541.55 Hz | 382.78 Hz | 206.30 Hz | |

586.53 Hz | 541.54 Hz | 382.78 Hz | 206.31 Hz |

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**Figure 2.**Stages from the basic disk section to the blade shape: (

**a**) Basic disk section, (

**b**) Transition 1, (

**c**) Transition 2, (

**d**) Transition 3, (

**e**) Final blade model, (

**f**) Blade main dimensions.

**Figure 3.**Torsional modes: (

**a**) Blades without a nodal diameter between them, (

**b**) blades with a nodal diameter between them.

**Figure 6.**Frequency response function of the Kaplan runner and the mode shapes extracted: (

**a**). 31.625 Hz, (

**b**). 35.875 Hz, (

**c**). 36.625 Hz, (

**d**). 39.375 Hz, (

**e**). 45.625 Hz, (

**f**). 49.625 Hz, (

**g**). 50.375 Hz, (

**h**). 54.125 Hz, (

**i**). 68.750 Hz, (

**j**). 87.625 Hz, (

**k**). 91.500 Hz.

D | 0 | 1 | 2 | 3 | … | m | |
---|---|---|---|---|---|---|---|

C | |||||||

0 | … | (m, 0) | |||||

1 | … | (m, 1) | |||||

$\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\vdots $ | $\ddots $ | $\vdots $ | |

n | (0, n) | (1, n) | (2, n) | (3, n) | … | (m, n) |

Disk | Transition 1 | Transition 2 | Transition 3 | Final Model |
---|---|---|---|---|

Disk | Z = 4 | Z = 5 | Z = 6 | Z = 7 |
---|---|---|---|---|

167.00 | 104.52 | 118.19 | 133.65 | 145.46 |

Disk | Z = 4 | Z = 5 | Z = 6 | Z = 7 |
---|---|---|---|---|

205.64 | 107.81 | 123.72 | 144.19 | 160.87 |

Disk | Transition 1 | Transition 2 | Transition 3 | Bladed Disk |
---|---|---|---|---|

270.9 Hz | 264.98 Hz | 245.03 Hz | 152.5 Hz | 101.26 Hz |

270.9 Hz | 266.55 Hz | 254.25 Hz | 216.49 Hz | 144.87 Hz |

Disk | Transition 1 | Transition 2 | Transition 3 | Bladed Disk |
---|---|---|---|---|

600.80 Hz | 583.52 Hz | 520.45 Hz | 313.16 | 162.84 Hz |

600.80 Hz | 583.99 Hz | 539.40 Hz | 430.09 Hz | 224.71 Hz |

Disk | Transition 1 | Transition 2 | Transition 3 | Bladed Disk |
---|---|---|---|---|

1050.00 Hz | 985.55 Hz | 778.21 Hz | 335.10 Hz | 160.50 Hz |

1050.00 Hz | 995.04 Hz | 896.11 Hz | 747.68 Hz | 652.96 Hz |

Disk | Transition 1 | Transition 2 | Transition 3 | Bladed Disk |
---|---|---|---|---|

2259.10 Hz | 2099.6 Hz | 1572.40 Hz | 772.65 Hz | 254.37 Hz |

2259.10 Hz | 2131.30 Hz | 1913.30 Hz | 1694.40 Hz | 1447.6 Hz |

Disk | Transition 1 | Transition 2 | Transition 3 | Bladed Disk |
---|---|---|---|---|

1604.30 Hz | 1532.00 Hz | 1261.80 Hz | 524.68 Hz | 200.53 Hz |

1604.30 | 1540.70 Hz | 1421.50 Hz | 1315.60 Hz | 1185.30 Hz |

Disk | Transition 1 | Transition 2 | Transition 3 | Bladed Disk |
---|---|---|---|---|

3010.90 Hz | 2797.8 Hz | 2107.70 Hz | 1060.60 Hz | 319.08 Hz |

3010.90 Hz | 2840.2 Hz | 2545.90 Hz | 2207.40 Hz | 1748.70 Hz |

Disk | Z = 4 | Z = 5 | Z = 6 | Z = 7 |
---|---|---|---|---|

270.9 Hz | 101.26 Hz | 120.32 Hz | 143.41 Hz | 163.58 Hz |

270.9 Hz | 144.78 Hz | 120.31 Hz | 143.41 Hz | 163.58 Hz |

Disk | Z = 4 | Z = 5 | Z = 6 | Z = 7 |
---|---|---|---|---|

600.80 Hz | 154.13 Hz | 177.96 Hz | 163.84 Hz | 206.3 Hz |

600.80 Hz | 154.12 Hz | 177.91 Hz | 224.71 Hz | 206.31 Hz |

Disk | Z = 4 | Z = 5 | Z = 6 | Z = 7 |
---|---|---|---|---|

1212.10 Hz | 1073.90 Hz | 745.33 Hz | 724.85 Hz | 721.68 Hz |

Mode | (2,0)1B | (2,0)1T | (4,0)1T | (4,0)2T |
---|---|---|---|---|

ND | 0 | 2 | 2 | 4 |

cPD | 2 | 0 | 2 | 0 |

Dimension | Symbol | Percentage of D_{o} [%] |
---|---|---|

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 |

Shaft Dominated Mode shape | |||||

5.852 Hz | 22.138 Hz | ||||

Generator Dominated Mode Shapes | |||||

28.654 Hz | 28.785 Hz | 41.959 Hz | 44.135 Hz | 44.234 Hz | |

Runner Dominated Mode Shapes | |||||

34.945 Hz | 38.443 Hz | 38.751 Hz | 44.298 Hz | 49.992 Hz | |

51.120 Hz | 51.269 Hz | 53.140 Hz | 53.444 Hz |

Mode | Bladed Disk | Kaplan Runner | Mode | Bladed Disk | Kaplan Runner |
---|---|---|---|---|---|

(0,0)1B | (3,0)1T-1 | ||||

(2,0)1B | (3,0)1T-2 | ||||

(1,0)1B | - | (4,0)1T-1 | - | ||

(2,0)1T | (4,0)1T-2 | ||||

(3,0)1T1B | - |

Mode | FEM model | Real Runner | Mode | FEM Model | Real Runner |
---|---|---|---|---|---|

(0,0)1B | (3,0)1T | Not extracted | |||

34.945 Hz | 39.375 Hz | 45.625 Hz | |||

(2,0)1B | (3,0)1T-1 | Not Detected | |||

38.443 Hz | 36.625 Hz | 51.120 Hz | |||

(1,0)1B | Not Detected | (3,0)1T-2 | Not Detected | ||

38.751 Hz | 53.444 Hz | ||||

(2,0)1T | (4,0)1T-1 | Not detected | |||

51.269 Hz | 49.625 Hz | 49.992 Hz | |||

(3,0)1T1B | Not Detected | (4,0)1T-2 | |||

44.298 Hz | 53.140 Hz | 50.375 Hz |

Mode | 1Tb1-1 | 1Tb1-2 | 2Tb1,2 | 2Tb1 |
---|---|---|---|---|

54.125 Hz | 68.750 Hz | 87.625 Hz | 91.500 Hz |

34.593 Hz | 37.977 Hz | 38.391 Hz | 43.913 Hz | 49.393 Hz |

50.498 Hz | 50.554 Hz | 52.438 Hz | 52.712 Hz |

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## Share and Cite

**MDPI and ACS Style**

Moraga, G.; Egusquiza, M.; Valentín, D.; Valero, C.; Presas, A.
Analysis of the Mode Shapes of Kaplan Runners. *Appl. Sci.* **2022**, *12*, 6708.
https://doi.org/10.3390/app12136708

**AMA Style**

Moraga G, Egusquiza M, Valentín D, Valero C, Presas A.
Analysis of the Mode Shapes of Kaplan Runners. *Applied Sciences*. 2022; 12(13):6708.
https://doi.org/10.3390/app12136708

**Chicago/Turabian Style**

Moraga, Greco, Mònica Egusquiza, David Valentín, Carme Valero, and Alexandre Presas.
2022. "Analysis of the Mode Shapes of Kaplan Runners" *Applied Sciences* 12, no. 13: 6708.
https://doi.org/10.3390/app12136708