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Analysis of a Linear Model for Non-Synchronous Vibrations Near Stall^{ †}

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## Abstract

**:**

## 1. Introduction

- The aeroelastic system is linearly unstable.
- The unsteady aerodynamic forces are generated by the blade vibration, i.e., the instability requires the participation of the structure.

- It occurs near the stall boundary but before rotating stall cells form.
- Prior to convective NSV, frequency spectra of unsteady pressure contain broadband frequencies, which result from multi-wave number disturbances propagating at approximately 50% of the rotor speed in the direction opposing rotation (in the rotor frame of reference).
- At the onset of vibration, the aerodynamic disturbance locks in with the structural vibration and the broadband spectrum changes to a coherent aerodynamic disturbance with a distinct frequency peak.

- The mean aerodynamics must promote the circumferential convection of vorticity, i.e., the blade must react to small changes in incidence by shedding vorticity, while the passage must be blocked close to the casing to allow the circumferential transport.
- The blade vibration must be able to modulate the propagation velocity of the aerodynamic disturbance to create a coherent disturbance in resonance with the vibration pattern.

## 2. Review of NSV Model

## 3. Frequency Domain Description

#### 3.1. Derivation of Influence Coefficients

#### 3.1.1. Before Lock-In

#### 3.1.2. After Lock-In

#### 3.2. Comparison to Classical AIC Approach

## 4. Rotor Stability Analysis

#### 4.1. Test Case

#### Validation of Frequency Model

#### 4.2. Application to Cases from Literature

## 5. Parametric Investigation

#### 5.1. Influence of Decay Rate

#### 5.2. Influence of Propagation Speed

#### 5.3. Influence of Amplitude and Phase of Aerodynamic Coefficients

## 6. Discussion

- The forcing due to multiple disturbances can be linearly superposed.
- The amplitude of the aerodynamic disturbance and therefore the forcing coefficient, ${\tilde{F}}_{a,0}$, depend linearly on vibration amplitude.
- The forcing coefficient ${\tilde{F}}_{a,0}$ does not change with blade oscillation phase.
- All blades are aerodynamically and structurally identical.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Sketch of rotor indicating blade numbering and sign convention and propagation of disturbance and (

**b**) schematic of the model.

**Figure 4.**Aerodynamic damping (

**a**) and aeroelastic frequency ratio (

**b**) versus nodal diameter for the experimentally tested operating points.

**Figure 5.**(

**a**) Critical (unstable) nodal diameter as function of propagation speed. Symbol size proportional to amplitude of $\zeta $. (

**b**) Damping coefficient for variation of propagation speed.

**Figure 7.**(

**a**) Variation of phase of ${\tilde{F}}_{v}$ and (

**b**) variation of phase of ${\tilde{F}}_{a}^{*}$.

OP1 | OP2 | |
---|---|---|

Decay rate, ${r}_{a}$ | $1.0$ | $0.075$ |

Propagation speed, ${\mathsf{\Omega}}_{aN}^{R}={\mathsf{\Omega}}_{a}^{R}/{\mathsf{\Omega}}_{r}^{R}$ | $-0.44$ | $-0.44$ |

Force coefficient ${\tilde{F}}_{a}^{*}={\tilde{C}}_{a}^{*}\widehat{\alpha}$ | $12{e}^{i(-0.50\pi )}$ | $12{e}^{i(-0.375\pi )}$ |

Force coefficient ${\tilde{F}}_{v}={\tilde{C}}_{v}\widehat{\alpha}$ | $50{e}^{i\left(0.25\pi \right)}$ | $50{e}^{i\left(0.25\pi \right)}$ |

Number of blades ${N}_{B}$ | 21 | 21 |

Vibration frequency ${\omega}_{v}^{R}/{\mathsf{\Omega}}_{r}$ | $5.81$ | $5.81$ |

**Table 2.**Cases from the literature successfully captured by the model, listing model inputs (number of blades, propagation speed and vibration frequency), experimentally measured nodal diameter ${N}_{v,exp}$ and nodal diameter predicted by the reduced order model ${N}_{v,model}$.

${\mathit{N}}_{\mathit{B}}$ | ${{\Omega}}_{\mathit{a}}^{\mathit{R}}/{{\Omega}}_{\mathit{r}}$ | ${\mathit{\omega}}_{\mathit{v}}/{{\Omega}}_{\mathit{r}}$ | ${\mathit{\sigma}}_{\mathit{v},\mathit{exp}}$ | ${\mathit{N}}_{\mathit{v},\mathit{exp}}$ | ${\mathit{N}}_{\mathit{v},\mathit{model}}$ | |
---|---|---|---|---|---|---|

Baumgartner et al. [9] | 27 | $-0.36$ | $7.56$ | ${80}^{\circ}$ | 6 | 6 |

Kameier and Neise [20] | 24 | $-0.6$ | $22.2$ | ${165}^{\circ}$ | 11 | 11 |

Brandstetter et al. [13] | 21 | $-0.44$ | $5.8$ | ${137}^{\circ}$ | 8 | 8 |

Rodrigues et al. [21] | 16 | $-0.42$ | $5.46$ | ${67.5}^{\circ}$ | 3 | 3 |

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**MDPI and ACS Style**

Brandstetter, C.; Stapelfeldt, S.
Analysis of a Linear Model for Non-Synchronous Vibrations Near Stall. *Int. J. Turbomach. Propuls. Power* **2021**, *6*, 26.
https://doi.org/10.3390/ijtpp6030026

**AMA Style**

Brandstetter C, Stapelfeldt S.
Analysis of a Linear Model for Non-Synchronous Vibrations Near Stall. *International Journal of Turbomachinery, Propulsion and Power*. 2021; 6(3):26.
https://doi.org/10.3390/ijtpp6030026

**Chicago/Turabian Style**

Brandstetter, Christoph, and Sina Stapelfeldt.
2021. "Analysis of a Linear Model for Non-Synchronous Vibrations Near Stall" *International Journal of Turbomachinery, Propulsion and Power* 6, no. 3: 26.
https://doi.org/10.3390/ijtpp6030026