#
Quantification of Blade Vibration Amplitude in Turbomachinery^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{7}loading cycles (without static loading) are performed with fabricated blades. Using Finite Element Method (FEM) calculations, it is possible to determine the maximum allowed stress for each blade eigenmode based on the Goodman diagram with the static loading at design conditions [6]. Based on modal solutions (displacement, strain, stress) from FEM calculations described in [2], blade vibration amplitudes given in micro-strain ($\mathsf{\mu}$m) can be related to blade stresses and presented relative to the critical stress (maximum allowed stress divided by safety factor), i.e., in percent scope limit.

- brief introduction of experimental setup and test procedure
- summary of the mathematical description of spectral analysis methods
- review of the influence of frequency resolution on FT and PSD and problem formulation
- sensitivity analysis based on artificial signals
- derivation and application of correction approach
- summary and propositon of guideline for vibration monitoring

## 2. Test Facility ECL-B3

#### 2.1. Instrumentation

#### 2.2. Test Procedure

## 3. Spectral Analysis Methodology

## 4. Problem Statement

^{2}/m

^{2}Hz and can hence not directly be compared to modal scope limits given in the unit micro-strain ($\mathsf{\mu}$m/m).

^{2}/Hz) by integrating PSD ${S}_{xx}$ over the frequency range and taking the square root of the result. In discrete cases, this is achieved by multiplying each density spectrum in Figure 6a with the corresponding frequency bin width $\Delta f$ to receive the RMS value for each frequency ${f}_{v}$. To facilitate comparison with FFT spectra shown in Figure 5a, calculated RMS values are multiplied with $\sqrt{2}$ to derive an approximation of peak amplitudes under the assumption of perfectly harmonic oscillation

## 5. Analysis of Artificial Signals

#### 5.1. Mass-Oscillator Model and System Identification

#### 5.2. Influence of Bin Width

#### 5.3. Influence of Modal Damping and Excitation

## 6. Correction Approach

## 7. Results and Discussion

- 1.
- Perform initial measurement at stable operating conditions, i.e., far away from the (expected) stability limit.
- 2.
- Determine amplitude evolution with different bin widths in post-processing and perform curve fit of Equation (20) to derive correction parameter a for each eigenmode of interest.
- 3.
- Apply correction according to Equation (21) with correction parameter a to correct spectral amplitude estimation depending on desired bin width and investigated eigenmode.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**ST-FFT-based amplitude of blade-mounted strain gauge of first three eigenmodes over time for different bin widths $\Delta f$.

**Figure 5.**(

**a**) FFT spectra of blade-mounted strain gauge for different bin widths $\Delta f$. (

**b**) Maximum FFT amplitude as a function of bin width $\Delta f$.

**Figure 6.**(

**a**) PSD spectrum for different bin widths $\Delta f$. (

**b**) RMS spectra based on PSD calculation for different bin widths $\Delta f$.

**Figure 8.**System identification: (

**a**) FFT spectrum with $\Delta f=4\phantom{\rule{0.166667em}{0ex}}$Hz with peak fits for eigenmodes 1, 2, and 3. (

**b**) Frequency, amplitude, and damping evolution over mass-flow rate from peak fits for eigenmode 2.

**Figure 9.**Normalized displacement signal $x\left(t\right)/{\widehat{x}}_{max}$ of SDOF mass oscillator for different simulated eigendmodes with parameters from Table 1.

**Figure 10.**FFT application to SDOF signal $x\left(t\right)$ for different eigenmodes: (

**a**) Peak spectra for different bin widths $\Delta f\phantom{\rule{0.166667em}{0ex}}$. (

**b**) Evolution of maximum spectral amplitude over bin width $\Delta f$.

**Figure 11.**Correction factor a: (

**a**) Curve fit procedure. (

**b**) Evolution over mass-flow rate ${\dot{m}}_{std}$.

**Figure 12.**Application of correction approach: (

**a**) Corrected FFT spectra of blade-mounted strain gauge for different bin widths $\Delta f$. (

**b**) Corrected spectral amplitude evolution over bin width $\Delta f$.

Parameter | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

${\omega}_{v}$ | 2$\pi \xb7264.8\phantom{\rule{0.166667em}{0ex}}$Hz | 2$\pi \xb7650.0\phantom{\rule{0.166667em}{0ex}}$Hz | 2$\pi \xb7843.6\phantom{\rule{0.166667em}{0ex}}$Hz |

${\mu}_{\zeta}$ | 0.057 | 0.020 | 0.022 |

${\sigma}_{\zeta}$ | 0.017 | 0.007 | 0.005 |

${\mu}_{{\omega}_{f}}$ | 2$\pi \xb7264.8\phantom{\rule{0.166667em}{0ex}}$Hz | 2$\pi \xb7650.0\phantom{\rule{0.166667em}{0ex}}$Hz | 2$\pi \xb7843.6\phantom{\rule{0.166667em}{0ex}}$Hz |

${\sigma}_{{\omega}_{f}}$ | 2$\pi \xb72.85\phantom{\rule{0.166667em}{0ex}}$Hz | 2$\pi \xb72.81\phantom{\rule{0.166667em}{0ex}}$Hz | 2$\pi \xb72.56\phantom{\rule{0.166667em}{0ex}}$Hz |

${\sigma}_{\widehat{f}}$ | 7% | 5% | 8% |

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## Share and Cite

**MDPI and ACS Style**

Schneider, A.P.; Paoletti, B.; Ottavy, X.; Brandstetter, C.
Quantification of Blade Vibration Amplitude in Turbomachinery. *Int. J. Turbomach. Propuls. Power* **2024**, *9*, 10.
https://doi.org/10.3390/ijtpp9010010

**AMA Style**

Schneider AP, Paoletti B, Ottavy X, Brandstetter C.
Quantification of Blade Vibration Amplitude in Turbomachinery. *International Journal of Turbomachinery, Propulsion and Power*. 2024; 9(1):10.
https://doi.org/10.3390/ijtpp9010010

**Chicago/Turabian Style**

Schneider, Alexandra P., Benoit Paoletti, Xavier Ottavy, and Christoph Brandstetter.
2024. "Quantification of Blade Vibration Amplitude in Turbomachinery" *International Journal of Turbomachinery, Propulsion and Power* 9, no. 1: 10.
https://doi.org/10.3390/ijtpp9010010