# Balancing of Flexible Rotors Supported on Fluid Film Bearings by Means of Influence Coefficients Calculated by the Numerical Assembly Technique

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Equations of Motion

#### 2.2. Speed Dependent Parameters of Fluid Film Bearings

#### 2.3. Boundary and Interface Conditions

#### 2.4. Homogeneous Solution

#### 2.5. Particular Solution

#### 2.6. Assembly and Solution Procedure

#### 2.7. Recursive Eigenvalue Search

#### 2.8. Influence Coefficient Method

## 3. Results

^{®}Core

^{TM}i7-8700 CPU with 3.2 GHz running on a Windows 10 operating system using MATLAB

^{TM}version R2019a.

#### 3.1. Rotor Bearing System

#### 3.2. Influence Coefficient Matrix

#### 3.3. Concentrated Unbalance

#### 3.4. Distributed Unbalance

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FEM | Finite Element Method |

FRF | Frequency Response Function |

NAT | Numerical Assembly Technique |

rpm | Rotations per minute |

## References

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**Figure 2.**Geometrical definitions of a fluid-film bearing (adapted from [29]).

**Figure 3.**Rotor-bearing system (adapted from [27]).

z | m | ${\mathbf{\Theta}}_{\mathit{t}}$ | ${\mathbf{\Theta}}_{\mathit{p}}$ | F | b | ${\mathit{d}}_{\mathit{i}}$ | $\mathit{\mu}$ | c |
---|---|---|---|---|---|---|---|---|

m | kg | ${\mathrm{kgm}}^{2}$ | ${\mathrm{kgm}}^{2}$ | N | m | m | $\frac{\mathrm{Ns}}{{\mathrm{m}}^{2}}$ | m |

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

0.13 | 0 | 0 | 0 | 431 | 0.04 | 0.02 | 0.032 | $8\times {10}^{-5}$ |

0.31 | 14.7 | 0.0412 | 0.0735 | 0 | 0 | 0 | 0 | 0 |

0.47 | 23 | 0.0966 | 0.0180 | 0 | 0 | 0 | 0 | 0 |

0.71 | 33 | 0.1960 | 0.3722 | 0 | 0 | 0 | 0 | 0 |

0.91 | 0 | 0 | 0 | 380 | 0.04 | 0.02 | 0.032 | $8\times {10}^{-5}$ |

1.0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

NAT | FEM | Relative Error |
---|---|---|

43.014 | 43.015 | $0.0031\%$ |

55.059 | 55.060 | $0.0018\%$ |

154.28 | 154.29 | $0.0031\%$ |

160.75 | 161.76 | $0.6275\%$ |

z | U | $\mathit{\beta}$ |
---|---|---|

m | kg m | rad |

0.31 | $2.8229\times {10}^{-4}$ | 0 |

0.47 | $5.5135\times {10}^{-4}$ | $\frac{\pi}{2}$ |

0.71 | $2.5994\times {10}^{-3}$ | $\pi $ |

Number of Elements | x-Direction | y-Direction |
---|---|---|

6 | $1.2965\times {10}^{-2}$ | $1.3422\times {10}^{-2}$ |

24 | $5.6796\times {10}^{-5}$ | $5.6677\times {10}^{-5}$ |

100 | $1.1306\times {10}^{-7}$ | $1.3352\times {10}^{-7}$ |

z | U | $\mathit{\beta}$ |
---|---|---|

m | kg m | rad |

0.31 | $2.8229\times {10}^{-4}$ | $\pi $ |

0.47 | $5.5135\times {10}^{-4}$ | −$\frac{\pi}{2}$ |

0.71 | $2.5994\times {10}^{-3}$ | 0 |

z | U | $\mathit{\beta}$ |
---|---|---|

m | kg m | rad |

0.31 | $3.0180\times {10}^{-4}$ | −3.0885 |

0.47 | $5.2940\times {10}^{-4}$ | −1.5469 |

0.71 | $2.6122\times {10}^{-3}$ | 0.0060 |

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**MDPI and ACS Style**

Quinz, G.; Klanner, M.; Ellermann, K. Balancing of Flexible Rotors Supported on Fluid Film Bearings by Means of Influence Coefficients Calculated by the Numerical Assembly Technique. *Energies* **2022**, *15*, 2009.
https://doi.org/10.3390/en15062009

**AMA Style**

Quinz G, Klanner M, Ellermann K. Balancing of Flexible Rotors Supported on Fluid Film Bearings by Means of Influence Coefficients Calculated by the Numerical Assembly Technique. *Energies*. 2022; 15(6):2009.
https://doi.org/10.3390/en15062009

**Chicago/Turabian Style**

Quinz, Georg, Michael Klanner, and Katrin Ellermann. 2022. "Balancing of Flexible Rotors Supported on Fluid Film Bearings by Means of Influence Coefficients Calculated by the Numerical Assembly Technique" *Energies* 15, no. 6: 2009.
https://doi.org/10.3390/en15062009