# Research on Vibration Suppression Method Based on Coaxial Stacking Measurement

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## Abstract

**:**

## 1. Introduction

## 2. The Centroid Transfer Model in the Rotation Coordinate System

_{x},w

_{y},w

_{z})T = $\frac{\mathit{l}}{\left|\mathit{l}\right|}$. The vector l and the rotation angle ∂ can be obtained by the change of the vector $O{O}_{v}^{o}$ in the coordinate system OXYZ and OX′Y′Z′, respectively. In the coordinate system OXYZ, the position vector of the measurement surface center of the k-th rotor after assembly can be expressed as follows:

^{T}, according to the coordinate conversion relationship, then,

_{k}, the unbalanced excitation force Q

^{r}of the k-th rotor in the coordinate system OX′Y′Z′ due to the offset of the center of mass is shown in Equation (7) [14]:

## 3. Finite Element Analysis Model of Single Rotor System

#### 3.1. Equation of Motion of Shaft Element

**q**

^{s}= [V

_{1}W

_{1}B

_{1}F

_{1}V

_{2}W

_{2}B

_{2}F

_{2}]

^{T}.

_{l}, l, φ

_{s}, I, A, and E denote the density, length, shear deformation coefficient, section moment of inertia, section area, and elastic modulus of the shaft element respectively.

#### 3.2. Equation of Motion of Disk (Rotor) Element

**q**

^{d}is the generalized displacement vector of the rotor,

**q**

^{d}= [V W B F]

^{T}. ${\mathit{Q}}^{d}$ is the external force vector on the disk.

_{d}, I

_{d}, and I

_{p}are the mass, diametral moment of inertia, and polar mass moment of inertia for the corresponding rigid disk, respectively.

#### 3.3. Equation of Motion of Bearing Element

**C**

^{b}is bearing damping matrix,

**K**

^{b}is bearing stiffness matrix, ${\mathit{Q}}^{b}$ is the external force vector of the bearing,

**q**

^{b}is the generalized displacement vector of the bearing, and

**q**

^{b}= [V W B F]

^{T}.

_{yz}= k

_{zy}= c

_{yz}= c

_{yz}= 0 and k

_{yy}= k

_{zz}, c

_{yy}= c

_{zz}.

#### 3.4. Equation of Motion of the Rotor System

**q**of the entire rotor system can be expressed as follows:

**M**is the mass assembly matrix, which contains the translational and rotary effects of the shaft element and the disk element. Just like

**M**,

**G**is the gyroscopic assembly matrix that considers the effects of the shaft element and the disk element.

**C**is the damping assembly matrix.

**K**is the stiffness assembly matrix that considers the stiffness effects of the shaft element and the disk element.

**Q**is the excitation matrix. To study the influence of the unbalanced excitation force caused by the deviation of the center of mass on the rotor system,

**Q**only considers the unbalanced excitation force after the multi-stage rotor is stacked and assembled.

#### 3.5. Unbalanced Response Solution

## 4. Simulation Analysis

^{7}N/m, the rear support rigidity is 1.6 × 10

^{7}N/m, and the working speed is 10,000 rpm.

_{r2}and the third-stage rotor installation phase θ

_{r3}at the working speed of the rotor system assembled with the three-stage rotor. Analyzing Figure 3 shows that, at the same speed, adjusting the installation phase of the rotors at all levels will change the vibration response. When the second-stage rotor installation phase θ

_{r2}is 195° and the third-stage rotor installation phase θ

_{r3}is 0°, the maximum vibration response amplitude is 27.1 μm; when θ

_{r2}is 180° and θ

_{r3}is 180°, the minimum vibration response amplitude is 2.9 μm; compared with the worst assembly, the vibration amplitude is optimized by 89% in the optimal assembly. Therefore, the optimal assembly phase of the rotor can be obtained through the dynamic analysis model of the unbalanced vibration response of the multi-stage rotor after assembly established in this paper, and the vibration suppression of the combined rotor can be achieved.

## 5. Experimental Verification

- (1)
- Air-bearing turntable is used to provide the rotary measurement datum. The radial and axial accuracies of air-bearing turntable are 80 nm.
- (2)
- Centering and tilt worktable is used for adjusting the eccentricity and tilt of base surface of the rotors to make the geometric axis coincident with the rotation axis. The minimum adjustments of displacement and angle are 0.2 μm and 0.2″, respectively.
- (3)
- The chuck is used to fix the rotor.
- (4)
- The inductive sensors are used to collect the radial and axial surface data of the rotors, of which the resolutions are 0.1 μm.
- (5)
- The displacement of the horizontal guide rail is 800 mm.
- (6)
- The displacement of the vertical guide rail is 2000 mm.
- (7)
- The turbine disk is the measured rotor.

## 6. Conclusions and Discussion

- (1)
- From the center of mass transfer model, it can be seen that the rotor vibration amplitude is affected by the installation phase of the rotor at all levels. The dynamic analysis model of the unbalanced vibration response of the single-rotor system established in this paper can directly reflect the corresponding relationship between the rotor installation phase and the vibration amplitude at different speeds.
- (2)
- According to the dynamic analysis model of the unbalanced vibration response of the single-rotor system, the optimal assembly phase of the multi-stage rotor can be obtained, and then the unbalanced excitation force of the multi-stage rotor can be adjusted to achieve the combined rotor vibration suppression.
- (3)
- When the speed is 3000 rpm, the uncertainty introduced by measurement repeatability u(x
_{1}) = 1.00 μm, the uncertainty introduced by the resolution of eddy current sensor u(x_{2}) = 0.01 μm, and the standard uncertainty is U = 2.00 μm (k = 2). - (4)
- It is verified by the vibration experiment of the rotor test piece that the optimal assembly phase obtained by the dynamic analysis model of the unbalanced vibration response of the single-rotor system can effectively suppress the vibration of the combined rotor.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 3.**Corresponding vibration amplitude diagrams at the rear support under different assembly phases.

**Figure 5.**Rotor geometric parameter measuring device: (

**a**) four rotors assembly; (

**b**) rotary measuring instrument.

Part Name | Front Axle | Compressor Disk | Turbine Disk | Rear Axle |
---|---|---|---|---|

Radius of upper end face (mm) | 43 | 189 | 100 | 66 |

Radius of lower end face (mm) | 189 | 100 | 66 | 40 |

Height (mm) | 560 | 245 | 325.5 | 124.5 |

Geometric eccentricity error in d_{x}^{o} (mm) | 0.0026 | 0.0011 | −0.0119 | −0.0012 |

Geometric eccentricity error in d_{y}^{o} (mm) | 0.0419 | 0.0051 | −0.0047 | 0.0009 |

Geometric eccentricity error in d_{z}^{o} (mm) | 0.0021 | 0.0015 | 0.0003 | 0.0029 |

Tilt error θx (″) | −1.5 | −2 | 0.7 | −7.7 |

Tilt error θy (″) | −1.8 | −2.3 | 0.6 | −12.8 |

Tilt angel θt (″) | 2.3 | 3.1 | 0.9 | 15 |

Lowest point of tilt θl (°) | 220 | 221 | 51 | 221 |

Part Name | Front Axle | Compressor Disk | Turbine Disk | Rear Axle |
---|---|---|---|---|

mass (g) | 27,850 | 19,278 | 53,278 | 5032 |

Mass eccentricity error d_{x}^{c} (mm) | 0.0042 | −0.0080 | −0.0042 | 0.0124 |

Mass eccentricity error d_{y}^{c} (mm) | −0.0039 | −0.0067 | 0.0041 | 0.0111 |

Mass eccentricity error d_{z}^{c} (mm) | 380 | 91 | 218 | 51 |

**Table 3.**Comparison of amplitude measurement results of rotor experimental parts under different assembly strategies.

3000 rpm | 6000 rpm | 9000 rpm | |
---|---|---|---|

Optimal assembly | 7.4 μm | 13.1 μm | 9.2 μm |

Direct assembly | 10.5 μm | 16.3 μm | 11.8 μm |

Optimization effect | 29.8% | 19.6% | 22.0% |

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**MDPI and ACS Style**

Sun, C.; Li, R.; Chen, Z.; Mei, Y.; Wang, X.; Li, C.; Liu, Y.
Research on Vibration Suppression Method Based on Coaxial Stacking Measurement. *Mathematics* **2021**, *9*, 1438.
https://doi.org/10.3390/math9121438

**AMA Style**

Sun C, Li R, Chen Z, Mei Y, Wang X, Li C, Liu Y.
Research on Vibration Suppression Method Based on Coaxial Stacking Measurement. *Mathematics*. 2021; 9(12):1438.
https://doi.org/10.3390/math9121438

**Chicago/Turabian Style**

Sun, Chuanzhi, Ruirui Li, Ze Chen, Yingjie Mei, Xiaoming Wang, Chengtian Li, and Yongmeng Liu.
2021. "Research on Vibration Suppression Method Based on Coaxial Stacking Measurement" *Mathematics* 9, no. 12: 1438.
https://doi.org/10.3390/math9121438