# Towards Accurate Prediction of Unbalance Response, Oil Whirl and Oil Whip of Flexible Rotors Supported by Hydrodynamic Bearings

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## Abstract

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## 1. Introduction

## 2. Bearing Models

_{i}and temperature T

_{i}on top of the bearing; see Figure 1. At the axial ends of the bearing, the oil flows out at ambient pressure. All three bearing models are based on the Reynolds equation, which assumes laminar flow, neglects fluid inertia effects and assumes constant pressure throughout the height of the oil film due to the small oil film thickness. The pressure distribution in a thin lubricating film with film thickness h, which supports a shaft with radius R at rotation speed Ω, is described by:

#### 2.1. Bearing Model 1: Isoviscous Short Bearings with Half-Sommerfeld Cavitation Conditions.

- the flow is dominated by the pressure gradients in the axial direction, and thus, the pressure gradient in circumferential direction can be ignored: $\frac{\partial p}{\partial x}\ll \frac{\partial p}{\partial z}$.
- the fluid viscosity does not vary with time or location.
- the shaft is always aligned with the bearing bore, i.e., shaft tilting is not included in the fluid film thickness function.
- the pressure distribution is dominated by the hydrodynamic pressure buildup; therefore, the hydrostatic pressure contribution of the supply is neglected.

#### 2.2. Bearing Model 2: Finite Length Journal Bearing with Gümbel Cavitation Conditions Including a Lumped Thermal Model

#### 2.3. Bearing Model 3: Finite Length Journal Bearing Including a Mass-Conservative Cavitation Algorithm, Non-Newtonian Fluid Description, Shaft Tilting Kinematics and a Distributed Thermal Model

#### 2.4. Summary of Bearing Models

## 3. Rotor-Bearing System A: A High Speed Laval Rotor on Plain Journal Bearings

#### 3.1. Rotor and Bearing Layout

#### 3.2. Rotor Model and Linear Rotodynamic Response

- traversal of a critical speed caused by the first shaft bending mode at ${\Omega}_{critical}=366$ Hz.
- above this critical speed, the first rigid body mode (a cylindrical mode, see Figure 6b) becomes linearly unstable at ${\Omega}_{\mathrm{whirl}}={\Omega}_{\mathrm{critical}}/2=183$ Hz.
- at rotation speeds above 40.000rpm, the whirl locks into the first shaft bending mode, resulting in a whip mode, which combines rigid body motion and bending motion.

#### 3.3. Non-Linear Time-Transient Analysis

- the extension of the thermal model to include the heating of the shaft, the bearing housing and the oil in the inlet channel, as depicted in Figure 4. Especially the effect of heating the oil in the inlet channel inside the bearing housing just upstream the bearing was found to be having a major impact on the effective film temperature.

## 4. Rotor-Bearing System B: An Asymmetric Rotor with Multiple Disks on Plain Journal Bearings

#### 4.1. Rotor and Bearing Layout

#### 4.2. Linear Analysis: Campbell Plot

#### 4.3. Non-Linear Time-Transient Analysis

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Numerical Values of Rotor-Bearing System A

Symbol | Description | Value | Unit | Symbol | Description | Value | Unit |
---|---|---|---|---|---|---|---|

${\mathrm{A}}_{bh\_cs}$ | Cross-sectional area bearing housing | 3e${}^{-4}$ | m${}^{2}$ | ${k}_{oil}$ | Thermal conductivity of oil | 0.145 | $\frac{W}{mK}$ |

${\mathrm{A}}_{bh\_o}$ | Outer surface area bearing housing | 6e${}^{-3}$ | m${}^{2}$ | ${k}_{shaft}$ | Thermal conductivity of shaft | 44 | $\frac{W}{mK}$ |

${H}_{bh}$ | Convection coefficient from bearing housing | 5 | $\frac{W}{{m}^{2}K}$ | ${\mathrm{L}}_{char\_bh}$ | Characteristic length of conductance in bearing housing | 1.14e${}^{-2}$ | m |

${k}_{air}$ | Thermal conductivity of air | 2.8e${}^{-2}$ | $\frac{W}{mK}$ | ${\nu}_{air}$ | Kinematic viscosity of air | 17e${}^{-6}$ | $\frac{{m}^{2}}{s}$ |

${k}_{bh}$ | Thermal conductivity bearing housing | 201 | $\frac{W}{mK}$ | ${\mathrm{c}}_{p\_oil}$ | Specific heat of oil | 2.1e${}^{3}$ | $\frac{J}{kgK}$ |

$P{r}_{air}$ | Prandtl number of air | 0.69 | − |

## Appendix B. Numerical Values of Rotor-Bearing System B

Symbol | Description | Value | Unit | Symbol | Description | Value | Unit |
---|---|---|---|---|---|---|---|

${\mathrm{A}}_{bh\_cs}$ | Cross-sectional area bearing housing | 6e${}^{-4}$ | m${}^{2}$ | ${k}_{oil}$ | Thermal conductivity of oil | 0.145 | $\frac{W}{mK}$ |

${\mathrm{A}}_{bh\_o}$ | Outer surface area bearing housing | 5e${}^{-2}$ | m${}^{2}$ | ${k}_{shaft}$ | Thermal conductivity of shaft | 44 | $\frac{W}{mK}$ |

${H}_{bh}$ | Convection coefficient from bearing housing | 5 | $\frac{W}{{m}^{2}K}$ | ${\mathrm{L}}_{char\_bh}$ | Characteristic length of conductance in bearing housing | 2.9e${}^{-2}$ | m |

${k}_{air}$ | Thermal conductivity of air | 2.8e${}^{-2}$ | $\frac{W}{mK}$ | ${\nu}_{air}$ | Kinematic viscosity of air | 17e${}^{-6}$ | $\frac{{m}^{2}}{s}$ |

${k}_{bh}$ | Thermal conductivity bearing housing | 44 | $\frac{W}{mK}$ | ${\mathrm{c}}_{p\_oil}$ | Specific heat of oil | 2.1e${}^{3}$ | $\frac{J}{kgK}$ |

$P{r}_{air}$ | Prandtl number of air | 0.69 | − |

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**Figure 1.**Layout of the plain journal bearing. The oil film thickness is exaggerated for visualization.

**Figure 2.**Fluid domain and boundary conditions for the unwrapped 2D bearing model: ambient pressure $p=0$ is set at the axial ends of the fluid domain, and inlet pressure $p={p}_{i}$ is set on the inlet channel. A periodic boundary condition connects sides $x=-\pi R$ and $x=\pi R$.

**Figure 3.**Cross-sectional schematic of the rotor-bearing system indicating the thermal nodes of the thermal network model. For each bearing, an individual thermal network model is created. Thermal interaction between bearings is neglected here for simplicity, but can be included by simply coupling thermal networks at node ${T}_{\mathrm{s}\_\mathrm{out}}$.

**Figure 4.**Thermal network model. Friction heat ${Q}_{\psi}$ is generated in the bearing and is subsequently transported to ambient air by means of convection and conduction paths. Each block represents a thermal resistance. Due to the relatively low oil flow rate, there is considerable heat transfer between the oil and the bearing housing in the inlet channel. Therefore, the oil temperature at the inlet of the bearing is close to ${T}_{\mathrm{bh}\_\mathrm{in}}$ instead of its externally-imposed temperature ${T}_{\mathrm{i}}$. The heat to warm up the oil flow ${\dot{m}}_{\mathrm{oil}}$ at the inlet channel is mostly supplied by the bearing housing, hence its branch from ${T}_{\mathrm{bh}\_\mathrm{in}}$ instead of a more intuitive branch from ${T}_{\mathrm{film}}$.

**Figure 5.**Layout of Rotor A: a Laval rotor with a central disk and two small measurement disks near the bearings.

**Figure 6.**Results of the eigenvalue analysis of the linearized rotor-bearing system. (

**a**) Campbell plot predicting the critical speed traversal at 22.000 rpm, the onset of whirl at 21.800 rpm and the transition to whip above 40.000 rpm; (

**b**) mode shapes. Top: near the critical speed; middle: in the oil whirl condition; bottom: in oil whip condition.

**Figure 7.**Waterfall plots of the rotor response: displacement of the measurement disk adjacent to the bearing. Oil feed pressure: 2.8 bar; oil feed temperature: 25 °C; unbalance on central disk: 250 mg·mm.

**Figure 9.**Layout of Rotor B. The rotor is driven by a motor, which is coupled to the rotor at the left-hand side. There is an unbalance load on the disk between the bearings.

**Figure 10.**Results of the eigenvalue analysis of the linearized rotor-bearing system. (

**a**) Campbell plot predicting critical speed traversals of forward modes at 1720 rpm and 4560 rpm. The onset of whirl is predicted at 2400 rpm followed by a gradual transition to whip above 3000 rpm. Above 4500 rpm, another whirl mode becomes unstable; (

**b**) Mode shape. Top: near the first critical speed; middle: second critical speed; bottom: in oil whip conditions.

**Figure 11.**Synchronous and sub-synchronous response of rotor-bearing System B. Oil feed pressure: 4 bar; oil feed temperature: 25 °C; unbalance on central disk: 250 mg·mm.

Rotor Parameters | Bearing Parameters | ||||||
---|---|---|---|---|---|---|---|

Symbol | Description | Value | Unit | Symbol | Description | Value | Unit |

${L}_{s}$ | Bearing span | 0.12 | m | C | Bearing clearance | 11e${}^{-6}$ | m |

${D}_{s}$ | Shaft diameter | 6e${}^{-3}$ | m | ${L}_{b}$ | Bearing length | 3.6e${}^{-3}$ | m |

${L}_{d}$ | Central disk length | 9e${}^{-3}$ | m | ${D}_{i}$ | Oil inlet diameter | 2e${}^{-3}$ | m |

${D}_{d}$ | Central disk diameter | 35e${}^{-3}$ | m | a | Viscosity-temperature parameter | 4.4e${}^{-4}$ | Pa·s |

${L}_{m}$ | Measurement disk length | 4e${}^{-3}$ | m | b | Viscosity-temperature parameter | 633 | ${}^{\circ}$C |

${D}_{m}$ | Measurement disk diameter | 12e${}^{-3}$ | m | c | Viscosity-temperature parameter | 88.6 | ${}^{\circ}$C |

${E}_{s}$ | Modulus of elasticity of shaft | 215 | GPa | m | Viscosity-shear rate parameter | 0.8 | - |

${\rho}_{s}$ | Density of shaft material | 7850 | $\frac{kg}{{m}^{3}}$ | r | Viscosity-shear rate parameter | 0.5 | - |

$m{r}_{unb}$ | Unbalance at center disk | 250 | mg·mm | K | Viscosity-shear rate parameter | 7.2e${}^{-7}$ | s |

ρ | Oil density | 855 | $\frac{kg}{{m}^{3}}$ |

Rotor Parameters | Bearing Parameters | ||||||
---|---|---|---|---|---|---|---|

Symbol | Description | Value | Unit | Symbol | Description | Value | Unit |

${L}_{1}$ | Length of shaft section | 0.12 | m | C | Bearing clearance | 17.5e${}^{-6}$ | m |

${L}_{2}$ | Length of shaft section | 0.40 | m | ${L}_{b}$ | Bearing length | 15e${}^{-3}$ | m |

${L}_{3}$ | Length of shaft section | 0.143 | m | ${D}_{i}$ | Oil inlet diameter | 3e${}^{-3}$ | m |

${L}_{4}$ | Length of shaft section | 0.05 | m | a | Viscosity-temperature parameter | 1.08e${}^{-3}$ | Pa·s |

${L}_{d}$ | Disk length | 15e${}^{-3}$ | m | b | Viscosity-temperature parameter | 324.3 | °C |

${D}_{s}$ | Shaft diameter | 25.4e${}^{-3}$ | m | c | Viscosity-temperature parameter | 52.51 | °C |

${D}_{d}$ | Disk diameter | 170e${}^{-3}$ | m | m | Viscosity-shear rate parameter | 0.8 | - |

${E}_{s}$ | Modulus of elasticity of shaft | 210 | GPa | r | Viscosity-shear rate parameter | 0.5 | - |

${\rho}_{s}$ | Density of shaft material | 7800 | $\frac{kg}{{m}^{3}}$ | K | Viscosity-shear rate parameter | 7.2e${}^{-7}$ | s |

$m{r}_{unb}$ | Unbalance on central disk | 189 | g·mm | ρ | Oil density | 879 | $\frac{kg}{{m}^{3}}$ |

Model 1 | Model 2 | Model 3 | |
---|---|---|---|

Pressure distribution | 1D | 2D | 2D |

Fluid supply hole | neglected | $p={p}_{i}\in $ inlet | $p={p}_{i}\in $ inlet |

Thermal distribution | Isoviscous | Lumped thermal | Distributed thermal |

Fluid type | Newtonian | Newtonian | Non-Newtonian |

Cavitation model | Half-Sommerfeld | Gümbel | Alakhramsing |

Shaft alignment (tilting) | Fully aligned | Fully aligned | Misalignment included |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Eling, R.; Te Wierik, M.; Van Ostayen, R.; Rixen, D.
Towards Accurate Prediction of Unbalance Response, Oil Whirl and Oil Whip of Flexible Rotors Supported by Hydrodynamic Bearings. *Lubricants* **2016**, *4*, 33.
https://doi.org/10.3390/lubricants4030033

**AMA Style**

Eling R, Te Wierik M, Van Ostayen R, Rixen D.
Towards Accurate Prediction of Unbalance Response, Oil Whirl and Oil Whip of Flexible Rotors Supported by Hydrodynamic Bearings. *Lubricants*. 2016; 4(3):33.
https://doi.org/10.3390/lubricants4030033

**Chicago/Turabian Style**

Eling, Rob, Mathys Te Wierik, Ron Van Ostayen, and Daniel Rixen.
2016. "Towards Accurate Prediction of Unbalance Response, Oil Whirl and Oil Whip of Flexible Rotors Supported by Hydrodynamic Bearings" *Lubricants* 4, no. 3: 33.
https://doi.org/10.3390/lubricants4030033