# Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Stability and Improved Breathing Function

_{c}is the area of closed portion. Assuming that stress concentration is not considered, the stress balance equation can be transformed by Hooke’s law as $\sigma =Ey/\rho $, where $y$ is the distance to neutral axis, E is Young’s modulus and ρ is bending radius. Since $E/\rho \ne 0$, the balance formula changes as $\int {}_{A}y\mathrm{d}A}=0$. As we all know, static moment is usually used to calculate the centroid of areas, i.e., ${x}_{A}={S}_{A}/A$. Thus, many researchers adopted the centroid formula to calculate the position of neutral axis. Hence, the distance from the neutral axis to point ${o}^{\prime}$ can be obtained as

_{2}, P

_{1}, P

_{0}are constant matrix calculated by

## 3. Simulation and Discussion

## 4. Conclusions

- The intervals of crack state are delayed by the initial bending, but the changes of interval size of each crack state are different. The size of fully open state is increased, the size of the fully closed one remains unchanged, and the two other interval sizes are reduced.
- The growth of crack reduces the natural frequency of the rotor system and increases the vibration amplitude, while the growth of the initial bending increases the amplitude and the range of resonance area.
- Many frequency components gradually appear as the crack grows, such as the natural frequency, combined frequency and multiple frequencies. On the contrary, initial bending gradually reduces these frequency components and makes them disappear. Stable regions are reduced, smoothed and extended laterally when rotating speed exceeds twice the natural frequency.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A cracked bearing-rotor system. (

**a**) structure schematic, (

**b**) cracked cross-section geometry.

**Figure 3.**Schematic diagrams of the cross-section. (

**a**) normal rotation position figure, (

**b**) adjusted position figure for the convenience of calculation.

**Figure 4.**Displacement of neutral axis and area of closed portion of the crack. (

**a**) neural axis, (

**b**) closed portion of the crack.

**Figure 5.**The moments of inertia about the rotating and fixed axes. (

**a**) about X axis, (

**b**) about Y axis.

**Figure 6.**Amplitude–speed diagram along with the growth of crack angle and initial bending. (

**a**) crack angle, (

**b**) initial bending as α = π/6.

**Figure 7.**Spectrogram of dynamic response affected by crack angle and initial bending. (

**a**) spectrogram with growth of crack, (

**b**) spectrogram with growth of initial bending.

Description | Value | Description | Value |
---|---|---|---|

Mass of bearing, ${m}_{1}$ | 1.0 kg | Mass of disk, ${m}_{2}$ | 32.1 kg |

Bearing radius, ${R}_{c}$ | 10 mm | Disk damping, ${c}_{2}$ | 2100 N·s/m |

Bearing Length, ${L}_{c}$ | 12 mm | Disk stiffness, $k$ | 2.5 × 10^{7} N/m |

Bearing damping, ${c}_{1}$ Friction coefficient, f | 1050 N·s/m 0.1 | Bearing stiffness, ${k}_{c}$ Radial clearance, c | 3.6 × 10^{6} N/m0.11 mm |

Shaft Length, L | 1 m | Shaft radius, R | 0.025 m |

$\mathit{\alpha}=\mathit{\pi}/6$ | Without Initial Bending | With Initial Bending | ||
---|---|---|---|---|

Ω = 200 (rad/s) | Ω = 500 (rad/s) | Ω = 800 (rad/s) | ||

Fully open | [−1.048, 1.048] | [−1.043, 1.078] | [−0.998, 1.122] | [−0.698, 1.423] |

From open to closed | [1.048, 2.094] | [1.078, 2.112] | [1.122, 2.156] | [1.423, 2.457] |

Fully closed | [2.094, 4.189] | [2.112, 4.206] | [2.156, 4.252] | [2.457, 4.552] |

From closed to open | [4.189, 5.236] | [4.206, 5.240] | [4.251, 5.285] | [4.552, 5.586] |

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**MDPI and ACS Style**

Wang, Y.; Xiong, X.; Hu, X.
Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending. *Machines* **2021**, *9*, 79.
https://doi.org/10.3390/machines9040079

**AMA Style**

Wang Y, Xiong X, Hu X.
Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending. *Machines*. 2021; 9(4):79.
https://doi.org/10.3390/machines9040079

**Chicago/Turabian Style**

Wang, Yuehua, Xin Xiong, and Xiong Hu.
2021. "Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending" *Machines* 9, no. 4: 79.
https://doi.org/10.3390/machines9040079