# Modal Balancing of Warped Rotors without Trial Runs Using the Numerical Assembly Technique

^{*}

## Abstract

**:**

## 1. Introduction

- Investigation of the optimal adaption of the model-based modal balancing procedure for warped rotors.
- Adaption of the procedure to measurements outside of critical speeds.
- Experimental verification of the proposed procedure.

## 2. Materials and Methods

#### 2.1. Numerical Assembly Technique

#### 2.1.1. Characteristic Equations

#### 2.1.2. Homogeneous Solution

#### 2.1.3. Boundary and Interface Conditions

#### 2.1.4. Particular Solution

#### 2.1.5. Assembly and Solution Procedure

#### 2.2. Modal Balancing Method

#### 2.3. Theory of Bow Compensation

## 3. Results and Discussion

#### 3.1. Test Bed

#### 3.2. Rotor Model

- Density: 7700 $\mathrm{kg}/{\mathrm{m}}^{3}$;
- Shear modulus: $8.1\times {10}^{11}\phantom{\rule{3.33333pt}{0ex}}\mathrm{N}/{\mathrm{m}}^{2}$;
- Shear correction factor: 0.89;
- External damping coefficient: 45 $\mathrm{Ns}/\mathrm{m}$.

- ${a}_{0}^{E}$: $2.2\times {10}^{11}$;
- ${a}_{1}^{E}$: $6.887\times {10}^{9}$;
- ${\alpha}^{E}$: $0.3$.

#### 3.3. Numerical Analysis

^{®}Core

^{TM}i7-8700 CPU running Windows 10 using MATLAB

^{TM}R2019a. For the numeric analysis, the rotor bow is neglected. The eigenvalues and Campbell diagram are found with a recursive search algorithm described in [32,33]. As is shown in Figure 6, the first two forward bending modes are calculated to be at 59.107 Hz and 145.48 Hz. These values will later be compared to measurements.

- Unknown parameters, especially the influence of the foundation and the coupling.
- Warped shafts.
- Measurement and mounting errors.

#### 3.4. Investigation of Optimal Bow Compensation

#### 3.5. Balancing of the First Mode of the Multi-Disk System

#### 3.6. Balancing of the First and Second Mode of the Multi-Disk System

#### 3.7. Discussion

- Despite the unknown foundation parameters, the eigenfrequencies were accurately predicted. In this test case, the shaft was substantially more flexible than the bearings and bearing foundations, reducing their influence. Further research could extend NAT with detailed foundation behaviour and investigate systems on elastic supports. Also, the balancing on rotors supported on fluid film bearings using NAT has only been theoretically presented [33], but has not yet been experimentally verified.
- The fractional time derivative damping model depicted the behaviour of the shaft correctly. Since the shaft is made out of steel, which shows very low viscoelastic properties, a less sophisticated material model would suffice. The applied material model in combination with NAT has already been used to calculate eigenfrequencies of strongly viscoelastic materials like Polyvinylchlorid [37], suggesting that the balancing of rotors made out of these materials could also be possible. This should be verified experimentally.
- Although NAT assumes a straight shaft, the vibration amplitudes of the first two modes of a warped rotor were successfully reduced using the presented procedure. Since the pre-bend of the test bed is significantly higher than the usually accepted values for high-speed machinery, it is inferred that a slight bow of the shaft poses no problem for the method.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FEM | Finite element method |

NAT | Numerical assembly technique |

rpm | Rotations per minute |

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**Figure 1.**General rotor problem [33].

**Figure 2.**Residual bow ${r}_{0}$, displacement due to unbalance excitation ${r}_{u}$ and total displacement ${r}_{I}$ of a warped rotor (inspired by [4]).

**Figure 3.**Ideal results of the balancing methods for a warped Laval rotor (inspired by [4]).

**Figure 4.**Rotordynamic test bed of the Institute of Mechanik of the University of Technology Graz without blast protection.

Left Disk | Right Disk | |
---|---|---|

Axial position | 88 mm | 448 mm |

Bow | 0.08694 mm | 0.11108 mm |

Angular position | ${173.91}^{\circ}$ | ${196.53}^{\circ}$ |

z | m | ${\mathbf{\Theta}}_{\mathit{xy}}$ | ${\mathbf{\Theta}}_{\mathit{zz}}$ | ${\mathit{c}}_{\mathit{x}}$ | ${\mathit{c}}_{\mathit{y}}$ | d |
---|---|---|---|---|---|---|

m | kg | kg m^{2} | kg m^{2} | N/m | N/m | Ns/m |

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

0.008 | 0 | 0 | 0 | 282,540,000 | 129,968,400 | 80 |

0.012 | 0 | 0 | 0 | 0 | 0 | 0 |

0.088 | 2.0200 | 0.00150 | 0.0028 | 0 | 0 | 0 |

0.143 | 0.1300 | $2.139\times {10}^{-5}$ | $3.4589\times {10}^{-5}$ | 0 | 0 | 0 |

0.268 | 0.0926 | $1.297\times {10}^{-5}$ | $1.9986\times {10}^{-5}$ | 0 | 0 | 0 |

0.448 | 2.0200 | 0.00150 | 0.0028 | 0 | 0 | 0 |

0.524 | 0 | 0 | 0 | 0 | 0 | 0 |

0.528 | 0 | 0 | 0 | 282,540,000 | 129,968,400 | 80 |

0.574 | 0.6471 | 0.0020322 | 0.00034134 | 0 | 0 | 0 |

0.596 | 0 | 0 | 0 | 0 | 0 | 0 |

Axial Position | Amount | Direction |
---|---|---|

0.088 m | 3.7233 $\times {10}^{-4}$ m | 4.6251 |

0.143 m | 1.600 $\times {10}^{-3}$ m | 3.1400 |

0.268 m | 1.600 $\times {10}^{-3}$ m | 4.800 |

0.448 m | 3.7233 $\times {10}^{-4}$ m | 5.0789 |

Left Disk | Right Disk | |
---|---|---|

Amount | 948 g mm | 1297 g mm |

Position | ${249.54}^{\circ}$ | ${283.97}^{\circ}$ |

Method 1 | Method 2 | Method 3 | |
---|---|---|---|

Amount | 8.71 g | 5.80 g | 7.63 g |

Position | ${5.720}^{\circ}$ | ${16.97}^{\circ}$ | ${9.224}^{\circ}$ |

Left Disk | Right Disk | |
---|---|---|

Amount | 1.4808 g | 9.0608 g |

Position | ${12.136}^{\circ}$ | ${359.82}^{\circ}$ |

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**MDPI and ACS Style**

Quinz, G.; Überwimmer, G.; Klanner, M.; Ellermann, K.
Modal Balancing of Warped Rotors without Trial Runs Using the Numerical Assembly Technique. *Machines* **2023**, *11*, 1073.
https://doi.org/10.3390/machines11121073

**AMA Style**

Quinz G, Überwimmer G, Klanner M, Ellermann K.
Modal Balancing of Warped Rotors without Trial Runs Using the Numerical Assembly Technique. *Machines*. 2023; 11(12):1073.
https://doi.org/10.3390/machines11121073

**Chicago/Turabian Style**

Quinz, Georg, Gregor Überwimmer, Michael Klanner, and Katrin Ellermann.
2023. "Modal Balancing of Warped Rotors without Trial Runs Using the Numerical Assembly Technique" *Machines* 11, no. 12: 1073.
https://doi.org/10.3390/machines11121073