# Influence of Aerodynamic Preloads and Clearance on the Dynamic Performance and Stability Characteristic of the Bump-Type Foil Air Bearing

^{*}

## Abstract

**:**

## 1. Introduction

#### On the Stability and Dynamic Performance of Lobed Bump-Type Foil Air Bearings

## 2. Bump-Type Foil Air Bearing Model

_{0}. The non-dimensional circumferential clearance ${C}_{0}$(θ

_{ij}), describing the distance between the journal and the foil at each circumferential node θ

_{ij}, represents the lubricant film at a centric journal position according to Equation (1) (with ${C}_{0}={c}_{0}$/${c}_{nom}$ and ${c}_{nom}$ = ${r}_{p}$ + ${c}_{m}$):

_{st}($\theta $ij) and damping c

_{st}($\theta $ij) = $\frac{\beta}{v}$k

_{st}($\theta $ij), based on the structural loss factor β and the frequency ratio v, can be established. Based on the structural stiffness k

_{st}and damping c

_{st}, the reaction force f

_{st}of the bump-foil can be obtained according to Equation (4):

## 3. Methods of Solution

#### 3.1. Steady-State Analysis

**VC**, see Table 1) are validated in Figure 3, using the results of Sim et al. [9]. Further numerical considerations on the FDM, boundary conditions and mathematical procedures are given in [16,20,21,22]:

_{sys}, using an eigenvalue analysis of the linear rotordynamic system according to Equation (11). In this way, the 4th order characteristic polynomial in Equation (12), expressed in a scalar product and based on the solution approach ${[x,y]}^{T}={[\hat{x},\hat{y}]}^{T}{e}^{\lambda t}$, requires a numerical solution. The system damping is represented by the real part and the eigenfrequency by the imaginary part of the computed eigenvalue (Equation (13)). Since the eigenvalue analysis is based on the linear dynamic coefficients, resulting from a linearization, the analysis is further denoted as linear stability analysis. Instability occurs at operational points that have zero or negative damping. Thus, excitation in the eigenfrequency consequently leads to a critical increase in the amplitude. Due to the lack of damping, entered energy can not be dissipated. Consequently, positive damping values of c

_{xx}and c

_{yy}dissipate energy, while Lund [25] emphasized that positive k

_{xy}values and negative k

_{yx}values provide negative system damping and supply energy to the journal motion. Referring to this, only the forward whirl leads to unstable motion, since this case provides negative damping [25]. Consequently, it can be summarized that cross-coupling stiffness and damping values represent a source of instability [29].

#### 3.2. Transient Analysis

_{0}. Starting from a defined eccentric point of the journal, the transient approach first calculates the corresponding steady-state quantities of foil deflection and pressure distribution until the minimum film thickness H

_{min}converges, as shown in Section 3.1. The resulting steady-state quantities of eccentricity, translocation velocity, foil deflection, and pressure are assigned as the initial values of the following transient analysis. Subsequently, the nondimensional time derivatives of the state system are calculated in each time step and are simultaneously solved using the explicit Adams–Bashfort method, a numerical solver for stiff differential equation systems [38]. Accordingly, each time-step calculates the instationary lubricant reaction forces, based on the integration of the instationary pressure distribution of the time-step according to Equation (5). The lubricant forces further allow the calculation of the derivative of the second and fourth state variable, the translocation acceleration. Solving the first order Reynolds equation, based on the pressure distribution of the current time-step, leads to the time derivative of the pressure $\dot{p}\left(\theta \right)$. Particularly, to calculate the foil deflection of the subsequent time-step, the corresponding time derivative $\dot{u}\left(\theta \right)$ needs to be calculated in the current time-step, which is derived from the formulation of the bump-foil reaction force in Equation (4).

## 4. Parametric Study

**TC 1**and

**TC 2**bearings, given in Table 1, are investigated. Within the parameter study, the preload factor is increased with a constant minimum clearance ${c}_{m}$ = 50 µm (

**TC 1**) and ${c}_{m}$ = 30 µm (

**TC 2**). In order to fully understand the influence of the aerodynamic preload on the dynamic performance and stability, this study covers a wide range of lobed bearing configurations from ${r}_{p}$/${c}_{m}$ = 0 to ${r}_{p}$/${c}_{m}$ = 12. Since the stiffness distribution of the foil is assumed to be uniform, the foil properties are kept constant according to the foil parameters in Table 1. Assuming a rotor weight of 1000 g, the static load on a single bearing satisfies 5 N. The rotor is in particular exposed to a harmonic, synchronous unbalance excitation, which is introduced with a constant value of $\rho $ = 1 gmm in the transient analysis. Regarding the steady-state analysis, the synchronous dynamic coefficients with a frequency ratio of v = $\omega $/$\mathsf{\Omega}$ = 1 consequently apply for the unbalance excitation case.

#### 4.1. Dynamic Performance

**TC 1**bearing shows a converging behavior, while the foil deflection of the

**TC 2**bearing decreases to a minimum and further increases at higher rotational speeds. Furthermore, the influence of the compliant bump foil directly affects the dynamic behavior of the bearing and and can be observed in the trend of the dynamic coefficients in Figure 5. In contrast to the stiffness ${k}_{yy}$ transverse to the load direction, the stiffness ${k}_{xx}$ in load direction at lower rotational speeds exceeds the stiffness at higher rotational speed as preload increases. As shown before, especially at low rotational speeds, the journal operates at high eccentricities, close to the minimum clearance circle and consequently at its loading limit. Due to the emerging lubricant reaction forces, the foil deflection increases decisively, raising the film thickness and stiffness. This characteristic dynamic behavior of the bump-foil highlights the promoting influence of the flexible structure, maintaining a carrying load capacity in limit load cases. More heavily lobed configurations exhibit a higher maximum foil deflection, forming a higher pressure peak in load direction and consequently a stronger focussed load distribution. Due to higher foil deflections, the dynamic behavior of the foil has a more significant contribution to the dynamic behavior of the bearing in stronger lobed configurations.

#### 4.2. Stability Analysis

**TC 1**. Both methods predict a decrease in journal eccentricity with increasing rotational speeds, caused by a stiffening of the lubricant film, indicating a good agreement. Since the transient analysis additionally considers the dynamic pressure, depending on the translocation velocity of the journal, the journal tends to lower eccentricities in the transient approach, compared to the steady-state method. Furthermore, the nonlinear method predicts an onset speed of instability of the ${r}_{p}$/${c}_{m}$ = 2 configuration at $\mathsf{\Omega}$ ≈ 105,000 rpm. The journal appears to perform an unstable motion, leading to critical minimum film thicknesses and causing the collapse of the lubricant film [38]. In terms of predicting the stationary eccentric position of the journal within the bearing, the linear and nonlinear methods agree well.

#### 4.3. Parameter Optimization

**TC 1**and

**TC 2**with a preload factor of ${r}_{p}$ = 300 µm, confirming a decisive reduction of the subsynchronous amplitude by decreasing the minimum clearance. In conclusion, to increase the stability characteristic of the bearing at the optimum lobe configuration, the minimum clearance must be minimized, which decisively increases the system damping. Further increasing the preload does not further increase the stability of the bearing and additionally degrades the load capacity and lift-off speed, as the stationary and transient prediction approaches have shown, which needs to be further experimentally investigated.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\left|A\right|$ | Amplitude [m] |

c | Damping [Ns/m] |

${c}_{0}/{C}_{0}$ | Circumferential bearing clearance [m]/[-] |

${c}_{nom}$ | Nominal bearing clearance [m] |

${c}_{m}$ | Minimum bearing clearance [-] |

${D}_{sys}$ | Damping ratio [-] |

D | Bearing diameter [m] |

E | Young’s modulus [Pa] |

$e/\u03f5$ | Eccentricity of shaft [m]/[-] |

${f}_{0}/{F}_{0}$ | Static load [N]/[-] |

f | Frequency [Hz] |

${f}_{b}/{F}_{b}$ | Lubricant reaction force [N]/[-] |

${f}_{st}$ | Bump foil reaction force [N] |

${f}_{max}$ | Maximum load capacity [N] |

${f}_{eig}$ | Eigenfrequency [Hz] |

${f}_{sub}$ | Subsynchronous frequency [Hz] |

$h/H$ | Film thickness [m]/[-] |

${h}_{min}/{H}_{min}$ | Minimum film thickness [m]/[-] |

k | Stiffness [Pa] |

L | Length of bearing [m] |

${l}_{0}$ | Half of bump length [m] |

${m}_{r}$ | Mass of the rotor [kg] |

n | Number of timesteps [-] |

${n}_{seg}$ | Number of foil segments [-] |

${n}_{b}$ | Number of bumps per segment [-] |

$p/P$ | Pressure [Pa]/[-] |

${p}_{a}$ | Ambient pressure [Pa] |

$\hat{Q}$ | Perturbed quantity [-] |

${R}_{b}$ | Bump radius [m] |

R | Bearing radius [m] |

${r}_{p}$ | Preload factor [m] |

s | Bump pitch [m] |

$t/\tau $ | Time [s]/[-] |

${t}_{b}$ | Foil thickness [m] |

$u/U$ | Foil deflection [m]/[-] |

v | Frequency ratio [-] |

$x/X$ | Vertical coordinate [m]/[-] |

$y/Y$ | Horizontal coordinate [m]/[-] |

$z/Z$ | Axial coordinate [m]/[-] |

${\alpha}_{0}$ | Static force angle [°] |

${\alpha}_{b}$ | Angle of lubricant reaction force [°] |

$\beta $ | Structural loss factor [-] |

$\gamma $ | Attitude angle of journal [°] |

$\theta $ | Circumferential coordinate [°] |

${\theta}_{p}$ | Pivot angle of foil segment [°] |

$\lambda $ | Eigenvalue [rad/s] |

$\mathsf{\Lambda}$ | Compressibility number [-] |

$\mu $ | Frictional coefficient [-] |

$\eta $ | Absolute viskosity [Pa s] |

$\nu $ | Poisson ratio [-] |

$\rho $ | Unbalance of shaft [g mm] |

$\psi $ | Nondimensional state variable [-] |

${\omega}_{0}$ | Undamped angular eigenfrequency [rad/s] |

$\omega $ | Excitation frequency [rad/s] |

$\mathsf{\Omega}$ | Rotational speed [1/min] |

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**Figure 1.**Bump-type foil air bearing model according to Sadri et al. [16].

**Figure 2.**Validation of the foil model under consideration of different load cases (LC) at a cantilever beam, set up by Le Lez et al. [17].

**Figure 3.**Validation of the steady-state eccentricity, minimum film thickness and attitude angle, based on results of Sim et al. [9], with respect to the

**VC**bearing with ${r}_{p}$ = 100 µm.

**Figure 4.**Validation of the synchronous dynamic coefficients, based on results of Sim et al. [9], with respect to the

**VC**bearing with ${r}_{p}$= 100 µm.

**Figure 5.**Synchronous dynamic coefficients of

**TC 1**as a function of ${r}_{p}$/${c}_{m}$ at different rotational speeds.

**Figure 6.**Maximum load capacity, eccentricity, minimum film thickness, and attitude angle as a function of ${r}_{p}$/${c}_{m}$ of

**TC 1**and

**TC 2**at different rotational speeds.

**Figure 7.**Pressure distribution of

**TC 1**and

**TC 2**in circumferential direction $\theta $ at $\mathsf{\Omega}$ = 60,000 rpm within the axial bearing center.

**Figure 8.**Maximum foil deflection of

**TC 1**and

**TC 2**in different lobe configurations as a function of rotational speed.

**Figure 9.**Steady-state linear and nonlinear eccentricity e and attitude angle $\gamma $ of the shaft of

**TC 1**as a function of rotational speed.

**Figure 10.**System damping ratio and eigenfrequency of the 1st and 2nd eigenmode of

**TC 1**of different lobed configurations as a function of rotational speed.

**Figure 11.**Numerically calculated run-up waterfall plots of

**TC 1**with different lobe configurations.

**Figure 12.**Amplitude and frequency of the 1st and 2nd subsynchronous response during rotor run-up of

**TC 1**with different lobed configurations.

**Figure 13.**System damping ratio and eigenfrequency of the 1st and 2nd eigenmode of

**TC 1**as a function of ${r}_{p}$/${c}_{m}$ at $\mathsf{\Omega}$ = 100,000 rpm.

**Table 1.**Bump-type foil air bearing properties of the test case

**TC 1**and

**TC 2**and the validation case bearings

**VC**according to [9].

Parameter | TC 1 | TC 2 | VC |
---|---|---|---|

Bore shape | |||

${c}_{m}$ [$\mathsf{\mu}$m] | 50 | 30 | 50 |

${r}_{p}$ [$\mathsf{\mu}$m] | 0 to 600 | 0 to 360 | 0, 100 |

D [mm] | 38.5 | 25.6 | |

L [mm] | 40 | 25.3 | |

Bump foil | |||

${n}_{seg}$ [-] | 3 | 3 | |

${\theta}_{p}$ [°] | 60 | 60 | |

${n}_{b}$ [-] | 9 | 6 | |

${R}_{b}$ [mm] | 2 | 2 | |

${l}_{0}$ [mm] | 1.81 | 1.2 | |

s [mm] | 4.57 | 2.7 | |

${t}_{b}$ [mm] | 0.127 | 0.12 | |

E [GPa] | 213 | 214 | |

$\nu $ [-] | 0.29 | 0.29 | |

$\mu $ [-] | 0.5 | 0.1 | |

$\beta $ [-] | 0.3 | 0.4 | |

Operational parameters | |||

${f}_{0}$ [N] | 5 | 1.9 | |

${\alpha}_{0}$ [°] | 0 | 0 | |

v (steady-state) [-] | 1 | 1 | |

$\rho $ (transient) [gmm] | 1 | - |

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**MDPI and ACS Style**

Walter, F.; Sinapius, M.
Influence of Aerodynamic Preloads and Clearance on the Dynamic Performance and Stability Characteristic of the Bump-Type Foil Air Bearing. *Machines* **2021**, *9*, 178.
https://doi.org/10.3390/machines9080178

**AMA Style**

Walter F, Sinapius M.
Influence of Aerodynamic Preloads and Clearance on the Dynamic Performance and Stability Characteristic of the Bump-Type Foil Air Bearing. *Machines*. 2021; 9(8):178.
https://doi.org/10.3390/machines9080178

**Chicago/Turabian Style**

Walter, Fabian, and Michael Sinapius.
2021. "Influence of Aerodynamic Preloads and Clearance on the Dynamic Performance and Stability Characteristic of the Bump-Type Foil Air Bearing" *Machines* 9, no. 8: 178.
https://doi.org/10.3390/machines9080178