# Bifurcation and Nonlinear Behavior Analysis of Dual-Directional Coupled Aerodynamic Bearing Systems

^{1}

^{2}

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## Abstract

**:**

_{r}) is 5.7 kg, chaotic regions where the maximum Lyapunov exponents are greater than 0 occur at bearing number ranges of 3.96–3.98 and 4.63–5.02. The coupling effect of the rotor mass and bearing number was also determined. This effect can provide an important guideline for avoiding an unstable state.

## 1. Introduction

## 2. Governing Equations of a DCAB System

#### 2.1. Design and Analysis of a DCAB System

_{x}, and θ

_{y}

_{,}) was designed. Moreover, the performance of single-thrust and radial bearings was compared. This comparison was conducted to determine the optimal design criteria for a coupled air-bearing system and to obtain a complete picture of the different motion behaviors of this system. The Reynolds equations of a coupled air-bearing system, including the lubrication equations of the thrust and radial bearings (presented in Equations (1) and (2), respectively), can be derived on the basis of narrow groove theory, [1,2]:

_{o}in Equations (5) and (6) represents the coordinates of the center of mass.

#### 2.2. Equations of Rotor Dynamics

_{r}supported by DCABs mounted on rigid pedestals. The rotor rotates with two degrees of oscillated displacement in the transverse plane. The equations of rotor dynamics for the transient state are presented in Cartesian coordinates in Equations (11) and (12). Moreover, the equilibrium equation of forces at the journal center is presented in Equation (13). In Equation (13), (x

_{2}, y

_{2}) and (x

_{3}, y

_{3}) are the displacements of the rotor and journal centers, respectively; ${\omega}_{r}$ is the rotor rotational speed; $\overline{\rho}$ is the rotor eccentricity; and ${\overline{K}}_{r}$ is the rotor stiffness coefficient.

## 3. Mathematical Formulation of the Numerical Simulation

## 4. Results

#### 4.1. Comparison of the Results Obtained with Different Numerical Methods

_{r}and Λ values.

#### 4.2. Dynamic Behavior of a DCAB System: Using the Rotor Mass as the Bifurcation Parameter

_{r}) is considered as the bifurcation parameter and the bearing number (Λ) of the system is considered to be 2.5.

#### 4.2.1. Dynamic Trajectory and Phase Plane Analysis

_{2}, Y

_{2}) has a regular trajectory for a low mass (m

_{r}= 11.2 kg). When the mass is increased to 20.92 kg, the regular motion changes to nonperiodic motion. When the rotor mass is maintained below 20.92 kg, the system is stable. The nonperiodic phenomenon becomes less obvious for rotor masses of 23.13, 24.31, 24.4, 24.52, 24.58, and 24.60 kg. When m

_{r}is 24.75 kg, the system suddenly becomes nonperiodic.

#### 4.2.2. Spectral Analysis

_{r}is 20.92 kg. When the rotor mass is increased to 23.13, 24.4, or 24.58 kg, the system motion becomes T-periodic. At m

_{r}values of 24.31, 24.52, and 24.60 kg, the system exhibits subharmonic 3T-periodic motion. Moreover, when the rotor mass is increased to 24.75 kg, the system displays chaotic motion.

#### 4.2.3. Bifurcation Analysis

_{r}on the DCAB system. The rotor masses are set between 0.1 and 25.0 kg in the operation of the DCAB system. Figure 4a,c indicates that the rotor in the system exhibits T-periodic motion in the horizontal and vertical directions when m

_{r}is less than 20.92 kg. This phenomenon is confirmed by analyzing the Poincaré section [Figure 6a]. When the rotor mass is increased to 20.92 kg, the T-periodic motion is replaced by quasi-periodic motion, as displayed in Figure 5b,d. The aforementioned state of motion is illustrated by the closed curve formed by the discrete points in the Poincaré section [Figure 6b]. The aforementioned motion is observed for an m

_{r}range of 20.92–23.13 kg. When the rotor mass is decreased to 20.13 kg, the system again exhibits T-periodic motion [Figure 6c]. This periodic motion is observed in the m

_{r}range of 20.13–24.31 kg. When the rotor mass is 24.31 kg, the original T-periodic motion bifurcates into 3T-periodic motion, as depicted in Figure 6d, in which three discrete points can be observed on the map. The 3T-periodic motion is observed within an m

_{r}range of 24.31–24.4 kg. The 3T-periodic motion changes to T-periodic motion when the rotor mass is increased to 24.4 kg, as displayed in Figure 6e. T-periodic motion is observed within an m

_{r}range of 24.4–24.52 kg. The aforementioned description indicates that the T- and 3T-periodic motions occur alternately over an m

_{r}range of 24.52–24.75 kg, as depicted in Figure 6f,h.

_{r}range of 24.75–25.0 kg. The types of rotor center behaviors for different rotor masses are presented in Table 4.

_{r}is 11.2, 20.92, 23.13, 24.31, 24.4, 24.52, 24.58, or 24.60 kg, the MLEs approach or are smaller than 0. Thus, the system exhibits nonchaotic behavior for the aforementioned m

_{r}values. When m

_{r}is 24.75 kg, as depicted in Figure 7i, the values of the MLEs are greater than 0; thus, the DCAB system exhibits chaotic motion. The MLE distribution of the DCAB system is displayed in Figure 8. Figure 8 indicates that the blue curve, which indicates chaotic motion (the MLEs are greater than 0), occurs between 24.75 and 25.0 kg. The red curve, which represents the stable situation, matches the aforementioned results and indicates that the rotor exhibits complicated types of motions.

#### 4.3. Dynamic Behavior of a DCAB System: Using the Bearing Number as the Bifurcation Parameter

_{r}is considered as 5.7 kg. The influence of Λ on the rotor dynamic behavior of the DCAB system is described in the following text.

#### 4.3.1. Dynamic Trajectory and Phase Plane Analysis

#### 4.3.2. Spectral Analysis

#### 4.3.3. Bifurcation Analysis

#### 4.4. Establishment of the Steady and Nonsteady Operation Regions of the DCAB System

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Dual-directional coupled aerodynamic bearing (DCAB): (

**a**) sectional view; (

**b**) coordinate system.

**Figure 3.**Dynamic trajectories (X

_{2},Y

_{2}) and phase plots (X

_{2},V

_{X2}) of the DCAB system for rotor masses of (

**a**) 11.2, (

**b**) 20.92, (

**c**) 23.13, (

**d**) 24.31, (

**e**) 24.4, (

**f**) 24.52, (

**g**) 24.58, (

**h**) 24.60, and (

**i**) 24.75 kg and a bearing number of 2.5.

**Figure 4.**Spectral responses along the horizontal and vertical directions of the DCAB system for rotor masses of (

**a**) 11.2, (

**b**) 20.92, (

**c**) 23.13, (

**d**) 24.31, (

**e**) 24.4, (

**f**) 24.52, (

**g**) 24.58, (

**h**) 24.60, and (

**i**) 24.75 kg and a bearing number of 2.5.

**Figure 5.**Bifurcation diagrams of the rotor center in the DCAB system for different masses and a bearing number of 2.5: (

**a**) X2(nT), (

**b**) partial X2(nT), (

**c**) Y2(nT), and (

**d**) partial Y2(nT).

**Figure 6.**Poincaré maps for different rotor masses: (

**a**) m

_{r}= 11.2 kg, (

**b**) m

_{r}= 20.92 kg, (

**c**) m

_{r}= 23.13 kg, (

**d**) m

_{r}= 24.31 kg, (

**e**) m

_{r}= 24.4 kg, (

**f**) m

_{r}= 24.52 kg, (

**g**) m

_{r}= 24.58 kg, (

**h**) m

_{r}= 24.60 kg, and (

**i**) m

_{r}= 24.75 kg.

**Figure 7.**Maximum Lyapunov exponents (MLEs) for different rotor masses: (

**a**) m

_{r}= 11.2 kg, (

**b**) m

_{r}= 20.92 kg, (

**c**) m

_{r}= 23.13 kg, (

**d**) m

_{r}= 24.31 kg, (

**e**) m

_{r}= 24.4 kg, (

**f**) m

_{r}= 24.52 kg, (

**g**) m

_{r}= 24.58 kg, (

**h**) m

_{r}= 24.60 kg, and (

**i**) m

_{r}= 24.75 kg.

**Figure 8.**Distribution of the MLEs for different rotor masses: (

**a**) 0.1 ≤ m

_{r}≤ 25.0 kg and (

**b**) 24.0 ≤ m

_{r}≤ 25.0 kg.

**Figure 9.**Dynamic trajectories(X

_{2},Y

_{2}) and phase plots(X

_{2},V

_{X2}) of the DCAB system for bearing numbers of (

**a**) 2.1, (

**b**) 3.21, (

**c**) 3.96, (

**d**) 3.98, (

**e**) 4.63, and (

**f**) 5.02 with a rotor mass of 4.2 kg.

**Figure 10.**Spectral responses along the horizontal and vertical directions of the DCAB system for bearing numbers of (a) 2.1, (

**b**) 3.21, (

**c**) 3.96, (

**d**) 3.98, (

**e**) 4.63, and (

**f**) 5.02 with a rotor mass of 4.2 kg.

**Figure 11.**Bifurcation diagrams for different bearing numbers when m

_{r}= 4.2 kg: (

**a**) X2(nT) and (

**b**) Y2(nT).

**Figure 12.**Poincaré maps for different bearing numbers: (

**a**) Λ = 2.1, (

**b**) Λ = 3.21, (

**c**) Λ = 3.96, (

**d**) Λ = 3.98, (

**e**) Λ = 4.63, and (

**f**) Λ = 5.02.

**Figure 13.**Variation of the MLEs with the number of driving force cycles for different bearing numbers: (

**a**) Λ = 2.1, (

**b**) Λ = 3.21, (

**c**) Λ = 3.96, (

**d**) Λ = 3.98, (

**e**) Λ = 4.63, and (

**f**) Λ = 5.02.

Variable | Meaning |
---|---|

x, y, z, ${\mathit{\theta}}_{\mathit{x}}$ and ${\mathit{\theta}}_{\mathit{y}}$ | degrees of freedom for five directions |

R | journal radius |

L | bearing length |

h | air film thickness |

P | air film pressure |

${\mathit{P}}_{\mathit{a}}$ | atmospheric pressure |

μ | viscosity coefficient |

r,θ,z | coordinate system |

t | time |

$\mathit{\xi}$ | physical quantity of perturbation |

${\widehat{\mathit{K}}}_{\mathit{J}},{\widehat{\mathit{K}}}_{\mathit{T}}$ | rigidity of journal bearing and thrust bearing, respectively |

${\widehat{\mathit{C}}}_{\mathit{J}},\text{}{\widehat{\mathit{C}}}_{\mathit{T}}$ | damping of journal bearing and thrust bearing, respectively |

m_{r} | rotor mass |

x_{2}, y_{2} | displacements of rotor center |

x_{3}, y_{3} | displacements of journal center |

${\mathit{\omega}}_{\mathit{r}}$ | rotational speed of rotor |

$\overline{\mathit{\rho}}$ | eccentricity of rotor |

${\overline{\mathit{K}}}_{\mathit{r}}$ | stiffness coefficient of rotor |

${\overline{\mathit{f}}}_{\mathit{a}\mathit{x}},\text{}{\overline{\mathit{f}}}_{\mathit{a}\mathit{y}}$ | equilibrium equation of forces acting to the journal center |

${\mathit{X}}_{\mathit{i}}$, ${\mathit{Y}}_{\mathit{i}}$, $\mathit{\tau}$, ${\overline{\mathit{F}}}_{\mathit{a}\mathit{x}}$,${\overline{\mathit{F}}}_{\mathit{a}\mathit{y}}$,${\mathit{M}}_{\mathit{r}}$, $\mathit{\zeta}$ | nondimensional group |

$\tilde{\mathit{H}}$ | time step |

Λ | dimensionless bearing number |

**Table 2.**Comparison of the displacements of the rotor center obtained with the three adopted numerical methods for different m

_{r}and Λ values.

Displacement | X2 | Y2 | |||
---|---|---|---|---|---|

Methods and Operating Conditions | $\tilde{\mathit{H}}={10}^{-3}$ | $\tilde{\mathit{H}}={10}^{-2}$ | $\tilde{\mathit{H}}={10}^{-3}$ | $\tilde{\mathit{H}}={10}^{-2}$ | |

Perturbation method | m_{r} = 11.2 kgΛ = 2.5 | −0.1823706379 | −0.1828876288 | 0.1788323797 | 0.1784390291 |

Traditional finite-difference method | −0.1827916378 | −0.1826400484 | 0.1785890294 | 0.1788268449 | |

Hybrid method | −0.1828351986 | −0.1828336582 | 0.1782085821 | 0.1782019995 | |

Perturbation method | m_{r} = 9.3 kgΛ = 2.5 | −0.1108152973 | −0.1100040131 | 0.0818502819 | 0.0810126263 |

Traditional finite-difference method | −0.1106937267 | −0.1101360701 | 0.0811265030 | 0.0813032587 | |

Hybrid method | −0.1105966888 | −0.1105993378 | 0.0812939956 | 0.0812912614 | |

Perturbation method | m_{r} = 5.7 kgΛ = 2.1 | 0.0743158143 | 0.0745814316 | −0.0022026672 | −0.0020779572 |

Traditional finite-difference method | 0.0742887199 | 0.0740141869 | −0.0023085543 | −0.0024181594 | |

Hybrid method | 0.07451222055 | 0.0745196645 | −0.0021489664 | −0.0021435547 | |

Perturbation method | m_{r} = 5.7 kgΛ = 3.0 | −0.2386678975 | −0.2384812317 | −0.0587582454 | −0.0582706309 |

Traditional finite-difference method | −0.2389323073 | −0.2387243330 | −0.0585839042 | −0.0587359340 | |

Hybrid method | −0.2385255159 | −0.2385242494 | −0.0584736366 | −0.0584706615 |

**Table 3.**Comparison of the values of the Poincaré sections of the rotor center in the DCAB system for different time intervals (τ; calculated using the hybrid method with m

_{r}= 5.6 kg and Λ = 2.5).

τ | X2(n T) | Y2(n T) |
---|---|---|

π/300 | −0.089320590874 | 0.052302554355 |

π/600 | −0.089329265030 | 0.052301040131 |

Rotor Mass | [0.1, 20.92) | [20.92, 23.13) | [23.13, 24.31) | [24.31, 24.4) | [24.4, 24.52) |

Dynamic Behavior | T | Quasi | T | 3T | T |

Rotor Mass | [24.52, 24.58) | [24.58, 24.6) | [24.6, 24.75) | [24.75, 25.0] | |

Dynamic Behavior | 3T | T | 3T | Chaotic |

**Table 5.**Dynamic behavior of the rotor center in the DCAB system for different bearing numbers (1.0 ≤ Λ ≤ 6.0).

Λ | [1.0, 3.21) | [3.21, 3.96) | [3.96, 3.98) |

Behavior | T | 2T | Chaos |

Λ | [3.98, 4.63) | [4.63, 5.02) | [5.02, 6.0] |

Behavior | 2T | Chaos | T |

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**MDPI and ACS Style**

Wang, C.-C.; Lin, C.-J.
Bifurcation and Nonlinear Behavior Analysis of Dual-Directional Coupled Aerodynamic Bearing Systems. *Symmetry* **2020**, *12*, 1521.
https://doi.org/10.3390/sym12091521

**AMA Style**

Wang C-C, Lin C-J.
Bifurcation and Nonlinear Behavior Analysis of Dual-Directional Coupled Aerodynamic Bearing Systems. *Symmetry*. 2020; 12(9):1521.
https://doi.org/10.3390/sym12091521

**Chicago/Turabian Style**

Wang, Cheng-Chi, and Chih-Jer Lin.
2020. "Bifurcation and Nonlinear Behavior Analysis of Dual-Directional Coupled Aerodynamic Bearing Systems" *Symmetry* 12, no. 9: 1521.
https://doi.org/10.3390/sym12091521