# Pulsation and Vibration Measurement on Stator Side for Turbocharger Turbine Blade Vibration Monitoring

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Review of Mengle’s Theory

- ω’: Observed frequency (in stationary frame)
- ω: Blade vibration frequency (in rotating frame)
- k: Traveling-wave mode number (in rotor structure)
- m: Integer (Fourier coefficient order for airborne wave between blades)
- B: Number of rotor blades
- Ω: Rotor rotational speed

- EO: Engine order,
- ND = k: Nodal diameter

#### 2.1.1. Rotor Structure-Borne Traveling Wave

#### 2.1.2. Airborne Wave Shape between Blades

#### 2.1.3. Pulsation Amplitude Relation with Blade Deflection

_{defl}can be formulated as

- ρ
_{0}: Density of medium - c
_{0}: Speed of sound - p
_{0}: Bulk gas static pressure - T
_{0}: Bulk gas temperature - τ ≤ 1: Fraction of gas flow travel time on the vibrating surface and one blade vibration cycle duration

_{defl}= 250 µm02P, atmospheric static pressure and a bulk gas temperature of 400 °C, the expected pulsation amplitude will be approximately 9 mbar02P at 2 kHz vibration frequency. When the vibration frequency is 10 kHz, it will be 43 mbar02P, i.e., the higher the frequency, the higher the pulsation amplitude that can be expected. In addition, at a higher vibration frequency, a higher τ can also be expected.

#### 2.1.4. Pulsation Amplitude Axial Decay Behaviour

**C**omputational

**F**luid

**D**ynamics (CFD) calculation for a one-stage axial turbine shows that the pulsation amplitude at blade-passing frequency will diminish to less than 1/20 of its original value approximately 12 times the chord length downstream from the turbine blade trailing edge. Therefore, also from this point of view, for an integral vibration mode detection, it is better to install a sensor away from the turbine blades.

_{x}, can be formulated as

- U: Axial bulk gas flow rate
- V: Circumferential bulk gas flow rate (= blade tip velocity)
- k
_{y}: Wave number between blades in circumferential direction

- s: Spacing between two blades

_{x}will have an imaginary part, so that the amplitude in the axial direction will decay. With a given set of U, V, ω, and c

_{0}, the content of the root will be a quadratic function of k

_{y}, which depends on the wave number between two blades, m.

^{2}+ V

^{2}– ${c}_{0}^{2}$ is always negative (relative velocity does not exceed the speed of sound), Y will be a convex parabolic curve.

#### 2.2. Experimental Setup

**F**inite

**E**lement

**M**ethod (FEM) calculation, so that the vibration behaviour of such resonances can be well characterized. Integrating an additional pulsation or accelerometer measurement into this already established process does not require too much extra effort. By this given process, it is possible to judge, during the turbine qualification period, if potentially critical resonances can be actually monitored by a sensor on the stator side. Pulsation, or sometimes also accelerometer measurements other than the abovementioned test series, have been conducted in the context of the standard HCF qualification tests.

## 3. Results and Discussion

**B**lade

**T**ip

**T**iming (BTT) program outputs, by default, a nodal diameter trace plot within

**L**east

**S**quares

**M**odel

**F**it (LSMF) analysis. From the plot, the dominant nodal diameter can be identified and, according to Equation (2), candidates for engine orders can be derived for the pulsation measurement analysis.

#### 3.1. Fundamental Vibration Mode Detection in a Radial Turbine

- (1)
- By BTT measurement and analysis, ND = −4 was identified as the dominant vibration mode;
- (2)
- By inserting ND = −4, EO = 6, B = 11 in (2), expected engine orders along which pulsation measurement reactions to the blade vibration mode will be processed as EO = 2 (m = 0), EO = 13 (m = 1), EO = 24 (m = 2) and so on;
- (3)
- By taking the aspect of airborne acoustic wave shape between two blades into account, the modes with m = 0 and m = 1 are expected to have high pulsation amplitudes;
- (4)
- Finally, by taking the axial amplitude decay behaviour into account (refer to Figure 3, the line with 3 kHz), only the mode with m = 1 (EO = 13) can propagate undiminished downstream (the pulsation sensors were located approx. 1.5 times the chord length downstream from turbine blade trailing edge).

#### 3.2. Fundamental Vibration Mode Detection in an Axial Turbine

_{radial}> τ

_{axial}.

#### 3.3. Nozzle-Ring-Induced Higher Vibration Mode Detection

#### 3.4. Long-Term Field Experience

## 4. Conclusions

## Funding

## Conflicts of Interest

## Nomenclature

A_{defl} | Blade deflection amplitude |

B | Blade number |

BTT | Blade Tip Timing |

CFD | Computational Fluid Dynamics |

EO | Engine or Excitation Order |

FEM | Finite Element Method |

FFT | First Fourier Transform |

HCF | High Cycle Fatigue |

HFO | Heavy Fuel Oil |

LSMF | Least Squares Model Fit |

ND | Nodal Diameter |

T_{0} | Bulk gas temperature |

U | Axial bulk gas flow rate |

V | Circumferential bulk gas flow rate (= blade tip velocity) |

Ω | Rotor rotational speed |

${c}_{0}$ | Speed of sound |

k | Traveling-wave mode number (in rotor structure) |

k_{x} | Airborne acoustic wave number in the axial direction |

k_{y} | Wave number between blades in circumferential direction |

m | Integer (Fourier coefficient order for airborne wave between blades) |

p | Pulsation amplitude |

s | Spacing between two blades |

ρ_{0} | Density of medium |

τ | Fraction of gas flow travel time on the vibrating surface and one blade vibration cycle duration |

ω | Blade vibration frequency (in rotating frame) |

ω’ | Observed frequency (in stationary frame) |

## References

- Kurkov, A. Flutter Spectral Measurements Using Stationary Pressure Transducers. J. Eng. Power.
**1981**, 103, 461–467. [Google Scholar] [CrossRef] [Green Version] - Kurkov, A.P. Formulation of Blade Flutter Spectral Analyses in Stationary Reference Frame; NASA TP-2296; NASA: Washington, DC, USA, March 1984. [Google Scholar]
- Mengle, V.G. Acoustic spectra and detection of vibrating rotor blades including row-to-row interference. Presented at the AIAA 13th Aeroacoustics Conference, Tallahassee, FL, USA, 22–24 October 1990. [Google Scholar]
- Gill, J.D.; Capece, V.R.; Fost, R.B. Experimental methods applied in a study of stall flutter in an axial flow fan. Shock. Vib.
**2004**, 11, 597–613. [Google Scholar] [CrossRef] [Green Version] - Murray, W.L., III; Key, N.L. Detection of Rotor Forced Response Vibrations Using Stationary Pressure Transducers in a Multistage Axial Compressor. Int. J. Rotating Mach.
**2015**, 2015, 198534. [Google Scholar] [CrossRef] - Agilis Measurement Systems. Available online: https://agilismeasurementsystems.com/ (accessed on 20 January 2020).

**Figure 2.**Schematic display of ND spatial distribution and its rotation speed in the case of EO = 6 and ND = −4, the resulting observed engine order from stator will be two.

**Figure 3.**Effect of blade vibration frequencies on the judgment if the airborne wave will propagate or decay in the axial direction.

**Figure 4.**Typical layout of turbocharger turbine stage with instrumentation ports, left: radial turbine; right: axial turbine.

**Figure 5.**Comparison between blade tip timing (BTT) ND -4 trace and pulsation amplitude trace on EO13 in a sweep test through a fundamental resonance mode on EO6 of a radial turbine with 11 blades.

**Figure 7.**Comparison between BTT ND +9 trace and pulsation amplitude trace on EO66 in a sweep test through a nozzle-ring-vane-induced higher vibration mode along EO24, with narrowest and widest nozzle ring throat configurations (axial turbine stage, B = 33).

**Figure 8.**Pulsation amplitude dependence on static pressure, >240 fundamental-mode resonance passages have been recorded over five months in a high-pressure radial turbine stage.

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ando, T.
Pulsation and Vibration Measurement on Stator Side for Turbocharger Turbine Blade Vibration Monitoring. *Int. J. Turbomach. Propuls. Power* **2020**, *5*, 11.
https://doi.org/10.3390/ijtpp5020011

**AMA Style**

Ando T.
Pulsation and Vibration Measurement on Stator Side for Turbocharger Turbine Blade Vibration Monitoring. *International Journal of Turbomachinery, Propulsion and Power*. 2020; 5(2):11.
https://doi.org/10.3390/ijtpp5020011

**Chicago/Turabian Style**

Ando, Takashi.
2020. "Pulsation and Vibration Measurement on Stator Side for Turbocharger Turbine Blade Vibration Monitoring" *International Journal of Turbomachinery, Propulsion and Power* 5, no. 2: 11.
https://doi.org/10.3390/ijtpp5020011