# Feasibility of Detecting Natural Frequencies of Hydraulic Turbines While in Operation, Using Strain Gauges

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Dynamics of a Francis Turbine Runner

#### 2.1. Modal Behavior of a Francis Turbine Runner

#### 2.2. Excitation Phenomena in Francis Turbines

#### 2.2.1. Periodic Excitation

_{v}is the number of guide vanes and Z

_{b}the number of runner blades; and m, n are integer numbers (1, 2, …, ∞) that represent the order of harmonic. The amplitude of the excitation A

_{m,n}depends on several parameters like the head (difference in altitude between the upper and the lower reservoir), operating point, design of the machine and order of harmonics (m, n). Therefore, RSI excitation is a sum of sinusoidal waves at different frequencies with different amplitudes. From the rotating frame or from the runner point of view, these frequencies are calculated as in Equation (2), where f

_{f}is the runner rotating frequency. The shape of the wave (k

_{m,n}) associated with each f

_{m}is a combination of the n harmonics of Z

_{b}and the m harmonics of Z

_{v}(Equation (3)).

_{m}with the shape k

_{m,n}. For example, for a combination of Z

_{b}= 16 runner blades and Z

_{v}= 20 guide vanes and a rotating speed of f

_{f}= 2.14 Hz (128.6 rpm), the main frequencies of excitation from the rotating frame are: f

_{1}= 42.8 Hz, f

_{2}= 85.6 Hz, f

_{3}= 128.4 Hz and f

_{4}= 171.2 Hz. The shape corresponding to each of those frequencies is given in Table 1, which is compiled according to Equation (2). The positive sign of the values in Table 1 means that the wave is rotating in the same direction as the runner, and the negative sign that it is rotating in the opposite direction. The shape of the excitation frequency f

_{1}corresponds to the first column of Table 1. In this case, it is a sum of a wave with 4 nodal diameters rotating in the opposite direction to the runner and waves with 12, 28, 44, 60, … nodal diameters rotating in the same direction as the runner. Only the smaller values of k

_{m,n}are usually considered because they are the ones that present higher amplitudes [31,32] (colored in red in Table 1). Therefore, f

_{1}is mainly a wave with 4 nodal diameters rotating in the opposite direction to the runner. In that way, f

_{2}is a −8, +8 wave, f

_{3}is a +4 wave and f

_{4}a 0 nodal diameters wave.

_{f}) increases from 0 to the nominal value in the case of the start-up and vice versa for the stop. This procedure of acceleration and deceleration of the runner can be of about 30 s of duration. This means that the excitation frequency of the RSI also changes from 0 to its steady value (f

_{m}) or vice versa (see Equation (2)). Therefore, if the natural frequency of the runner is below the steady value of the excitation frequency (f

_{m}) and the excitation shape coincides with the mode shape of the runner, a resonance is expected to happen. In this case, this natural frequency may be detected every start-up or stop. For instance, taking the values of the same example presented before (Z

_{b}= 16, Z

_{v}= 20 and f

_{f}= 2.14 Hz), if the natural frequency of the runner associated to a 4ND mode-shape is below f

_{1}= 42.8 Hz, this will be excited during the start-up and stop of the machine. The same applies for the rest of the excitation frequencies (f

_{2}, f

_{3}, f

_{4}, …, f

_{∞}) and the rest of the natural frequencies of the runner.

_{f}). In this case, circumferential pressure pulsations are generated at this low frequency, displacing the runner in the radial direction. For the overload vortex rope, the vortex is axially centered in the draft tube cone, inducing an axial force over the runner. The frequency of this overload vortex rope is in the same range as the part-load vortex rope. As this phenomenon occurs at a very low frequency, normally it does not match any natural frequency of the runner.

_{f}).

#### 2.2.2. Transient Excitation

#### 2.2.3. Random Excitation

## 3. Experimental Tests

#### 3.1. Description of the Selected Turbine

_{s}) of 46 (further information about hydraulic turbines’ specific speed can be found in [20]) and a rated power of 444 MW. The study of the dynamic behavior of this Francis Turbine is part of the collaborative European Project Hyperbole (FP7-ENERGY-2013-1) [44]. The runner has 16 blades (Z

_{b}= 16), whereas the distributor has 20 guide vanes (Z

_{v}= 20). The rotating speed of the machine is 128.6 rpm (2.14 Hz). Taking advantage of an overhaul in the power plant, the machine was accessible enough to install several sensors in the rotating parts and in the stationary parts. The objective of these tests was to identify all the phenomena that occur in every operating point and how they are detected with the different sensors.

#### 3.2. Instrumentation

#### 3.3. Testing Procedure

- Start-up (duration of 1 min): the machine goes from standstill to the nominal speed (128.6 rpm).
- Speed no load (SNL): the machine turns at nominal speed but without generating energy (no excitation in the generator). Low periodic excitation due to RSI and important random excitation due to turbulence.
- Deep part load (DPL): very low flow, so the machine was generating between 5% and 30% the rated power. Low periodic excitation due to RSI and important random excitation due to turbulence.
- Part load (PL): between 30% and 70% of the rated power. Medium periodic excitation of the RSI and high periodic excitation due to part-load vortex rope.
- Best efficiency point (BEP): for the head during the test, this point was at 90% of the rated power. High periodic excitation due to the RSI.
- Overload (OL): Between 100% and 108% of the rated power. High periodic excitation due to the RSI and due to the overload vortex rope.

## 4. Signal Processing

#### 4.1. Strain Modal Testing

#### 4.2. Joint Time–Frequency Analysis

#### 4.3. Averaged–Spectrum Analsysis

## 5. Results

#### 5.1. Natural Frequencies in Air

#### 5.2. Natural Frequencies under Operating Conditions

#### 5.2.1. Start-Up

_{1}). The second and third excitation lines due to the RSI are highlighted using a dotted red line (see Figure 9). During this acceleration, a frequency of 27 Hz is excited with high amplitude with the first excitation line (marked with a circle in the figure) and another frequency of 75 Hz is also excited with the third excitation line (again marked with a circle in the figure). These two frequencies seem to correspond, respectively, to those found at 25 Hz and 70 Hz in the beginning of the transient. The slight increase of these natural frequencies could be due to different amounts of cavitation between the initial hit and the acceleration of the machine, or because the natural frequencies of the runner are also affected by the rotating speed of the machine, as Presas et al. found in their experimental study [12].

#### 5.2.2. Deep Part Load

_{3}= 128.4 Hz), which according to Table 1 (column 3) is mainly a k

_{n.m}= 4 excitation; so it seems that it is more excited than other frequencies because it may be a 4ND mode shape. According to the natural frequencies in air and previous knowledge about natural frequencies in water of Francis turbines [10,22,23,24], this natural frequency (120–123 Hz) may correspond to the mode shape called “4ND-CD-IPh” in Table 2. This means that this mode shape has a reduction of 52% of the natural frequency in air when in operating conditions.

#### 5.2.3. Overload

_{vortex rope}) fluctuations due to the vortex rope can be observed clearly in the time plot. Every maximum of these fluctuations correspond to a strong shake of the runner, exciting all its natural frequencies. Different frequencies are excited during this moment: 28 Hz, 35 Hz, 75 Hz, 90–100 Hz and 120–123 Hz.

## 6. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Pictures of the Francis turbine runner studied. (

**a**) Picture during the installation of the strain gauges in the runner; (

**b**) detail of the runner blades.

**Figure 2.**Detail of the strain gauges installed in the runner. (

**a**) Strain gauges in the pressure side and band side; (

**b**) strain gauges in the suction side and crown side; (

**c**,

**d**) detail of the epoxy resin covering the strain gauges.

**Figure 3.**Detail of the telemetry system installed in the rotating train of the Francis turbine. (

**a**) Sketch of the telemetry system; (

**b**) picture of the telemetry system installed in the hub of the runner; (

**c**) rotating antenna at the tip of the shaft.

**Figure 4.**Detail of sensors available during the experimental modal analysis (EMA). (

**a**) Sketch of the location of the accelerometers; (

**b**) picture of the accelerometer located in the band outlet.

**Figure 5.**Time signal of the wicket gate opening, rotating speed and a strain gauge of the runner during the start-up of the machine.

**Figure 6.**Comparison of an averaged-spectrum analysis (

**left**) and a joint time–frequency analysis using wavelets (

**right**) of a strain gauge signal during deep part load (DPL) operation.

**Figure 7.**Frequency response function (FRF) in different points of the runner (0–250 Hz), as well as the operational deflection shape (ODS) of every runner mode shape associated with every natural frequency. Maximum displacement is colored red, and minimum displacement is colored blue. Videos of the mode shapes are found the Supplementary Materials section (S1 to S12).

**Figure 8.**FRFs (only the real part) of different strain gauges with the hammer as a reference (impact in blade 7).

**Figure 9.**Joint time–frequency plot of a strain-gauge signal (crown side, pressure side, blade 7) during the initial hit.

**Figure 10.**Joint time–frequency plot of a strain gauge signal (crown side, pressure side, blade 7) during DPL (18% of rated power).

**Figure 11.**Averaged-spectrum analysis of a strain-gauge signal (crown side, pressure side, blade 7) during (

**a**) DPL (18% of rated power); and (

**b**) best efficiency point (BEP) (90% of rated power).

**Figure 13.**Joint time–frequency plot of a strain-gauge signal (crown side, pressure side, blade 7) during overload (OL) condition (107% of rated power).

**Table 1.**Excitation shape (k

_{m,n}) table for a combination of Z

_{b}= 16 and Z

_{v}= 20 runner blades. The most important components are colored in red.

mZv/nZb | 20 (f_{1} = 42.8 Hz) | 40 (f_{2} = 85.6 Hz) | 60 (f_{3} = 128.4 Hz) | 80 (f_{4} = 171.2 Hz) |
---|---|---|---|---|

16 | −4 | −24 | −44 | −64 |

32 | 12 | −8 | −28 | −48 |

48 | 28 | 8 | −12 | −32 |

64 | 44 | 24 | 4 | −16 |

80 | 60 | 40 | 20 | 0 |

**Table 2.**First 12 mode shapes, natural frequencies and damping ratios of the runner in air. Results obtained by means of the EMA.

Mode-Shape Number | Mode-Shape Name | f_{n} [Hz] | Damping Ratio [%] |
---|---|---|---|

1 | 2ND-G-IPh | 46.44 | 1.0272 |

2 | 3ND-G-IPh | 98.24 | 0.5498 |

3 | 1ND-G-IPh | 129.14 | 2.4669 |

4 | 4ND-G-CPh | 148.08 | 0.3094 |

5 | 2ND-BlD-IPh | 155.58 | 0.3469 |

6 | 5ND-BlD-CPh | 181.03 | 0.2713 |

7 | 3ND-BlD-IPh | 192.48 | 0.2394 |

8 | 6ND-BlD-CPh | 197.89 | 0.2894 |

9 | 0ND-BlD-IPh | 206.85 | 0.2490 |

10 | 1ND-BlD-CPh | 209.38 | 0.2413 |

11 | 1ND-BlD-IPh | 216.99 | 0.8927 |

12 | 4ND-CD-IPh | 233.25 | 0.2550 |

Frequency Detected in Operation (Hz) | Start-Up | DPL | OL | Mode Shape | f_{air} (Hz) | f_{water}/f_{air} |
---|---|---|---|---|---|---|

28 | Yes | Yes | Yes | 2ND-G-IPh | 46.44 | 0.6029 |

35 | Yes | No | Yes | - | - | |

75 | Yes | No | Yes | 4ND-G-CPh | 148.08 | 0.5065 |

100 | Yes | No | Yes | - | - | |

123 | Yes | Yes | Yes | 4ND-CD-IPh | 233.25 | 0.5273 |

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## Share and Cite

**MDPI and ACS Style**

Valentín, D.; Presas, A.; Bossio, M.; Egusquiza, M.; Egusquiza, E.; Valero, C.
Feasibility of Detecting Natural Frequencies of Hydraulic Turbines While in Operation, Using Strain Gauges. *Sensors* **2018**, *18*, 174.
https://doi.org/10.3390/s18010174

**AMA Style**

Valentín D, Presas A, Bossio M, Egusquiza M, Egusquiza E, Valero C.
Feasibility of Detecting Natural Frequencies of Hydraulic Turbines While in Operation, Using Strain Gauges. *Sensors*. 2018; 18(1):174.
https://doi.org/10.3390/s18010174

**Chicago/Turabian Style**

Valentín, David, Alexandre Presas, Matias Bossio, Mònica Egusquiza, Eduard Egusquiza, and Carme Valero.
2018. "Feasibility of Detecting Natural Frequencies of Hydraulic Turbines While in Operation, Using Strain Gauges" *Sensors* 18, no. 1: 174.
https://doi.org/10.3390/s18010174