Comparison study regarding bearing performance on 3 types of air bearingsusing Dyrobes software

The gas bearings have low load capacity and high stiffness requirements beside the high precision which all are a concern in their development. The 2 large categories of gas bearings have each of them a drawback: the static gas bearing have the performances depending by the system which maintain the pressure, while the dynamic one's performances are increasing with speed. The article creates a comparison between 3 different type of dynamic gas bearing, using air as lubricant. While all 3 types of bearing having similar performances at max speed in term of eccentricity ratio vs. rpm, minimum film thickness and equilibrium locus, the cross-coupled damping and stiffness shown a rotor destabilization tendency. The 3 lobe rotor presented the highest pressure profiles at every speed. The pressure profiles present same shapes after 80000 rpm. The results shown potentially a multilobe rotor can over perform the plain bearing also in low speed.


Introduction
Initially the gas bearing found application where are required high speed, low load, oil and dust free environment like medical application but recently they also being applied successfully for hydrogen fuel cells in passenger cars [1,2]. The main concerns regarding gas bearings are high precision, very limited load, low performance at low speed for dynamic gas bearings, stiffness [1,3]. Despite those drawbacks, this type of bearing has some major advantages: oil or grease free, low viscosity, self-centering, low unbalance and they perform better where oil and ball contact bearings failing: at speeds over 80krpm [1,4].

Abbreviations
CAD -computer aided design FEA -finite element analysis

Simulation conditions
The length of bearing was considered L=80 mm, diameter D=10 mm, bearing clearance c=0,009 mm, ambient pressure P=1.01325 bar, load force W=11.7678 N and the gas dynamic viscosity µ=0.01837 centiPoise. The initial speed was considered at 5000 rpm and max speed 305000 rpm with an increment of 50000 rpm.
The bearing was considered a non-pressurized one with ambient side pressures P=1.01325 bar. The bearing characteristics were calculated using the following parameters:  Bearing characteristics number=µN/P  Where µ-absolute viscosity of lubricant (kg/m.s)  N-speed of the journal (rpm)  P-bearing pressure (N/mm^2)  In our case, for initial phase, considering the rotor speed:

Sommersfeld Number
The bearing characteristic number or Sommerfeld number (S, equation 1) is a dimensionless value, generally containing all the variables specified in bearing design and is used as a reference in hydrodynamic fluid film bearings analysis. [5]

( )
Where: S -Sommerfeld number (dimensionless) r -shaft radius (m) c -radial clearance (m) μ -absolute viscosity of the lubricant (pascal*s) ω -angular velocity of the shaft (rad/s) L -bearing length (m) D -bearing diameter (m) W -applied load (N) For all 3 cases Sommerfeld number is the same S=2018.168746 for 5000 rpm and S=100908.437281 for 250000 rpm.

Petroff's Law -shear stress in the lubricant
Bearing friction was first explained by Petroff's method (equation 2), who consider the shaft being concentric with bearing, and this method generates the equation called Petroff's Law from which results the friction coefficient. [6] τ -shear stress in the lubricant assuming a constant rate of shear. (pascal) π -pi r -shaft radius (m) μ -absolute viscosity of the lubricant (pascal*s) N -rotational speed of the shaft (revs/s) (1/s) c -radial clearance (m) The shear stress in all 3 air bearings: τ= 53435.9501872 at 5000 rpm; while at 250000 rpm τ= 2670453.660976444 Pascal.

Petroff's Law -Bearing coefficient of friction
Applicable to small load, Petroff's Law offers a reasonable estimation of friction coefficient [7]: Where: f -bearing coefficient of friction (dimensionless) π -pi μ -absolute viscosity of the lubricant (pascal*s) N -rotational speed of the shaft (revs/s) (1/s) P -radial load per unit of project bearing area (pressure) (Pascal) ( fig. 1) r -shaft radius (m) c -radial clearance (m) Fig. 1. Surface area was simulated and measured using Catia V5 CAD software.
The area was used to calculate the pressure of radial load. In all 3 bearings the coefficient to friction is: f= 0.0628634804849 for 5000 rpm and f=3.1431865467 for 250krpm.

Calculation and data evaluation
Calculation was performed using Dyrobes Bearing Performance module with Metric units. Eccentricity is decreasing with speed increase, due to self-centering property of air bearing. Having a more constant eccentricity results in also a more stabilized film thickness at higher speed. All three bearings performing quite similar regarding film thickness ( figure 6). In term of journal equilibrium locus, the lowest performance at 5000 rpm are the 3 lobe bearing, but all performing quite similar near the max speed ( figure 7).
At low rpm the 4 lobe and plain bearing pressures are close, only the 3 lobe bearing having a higher pressure (figure 8).
At 105000 rpm the pressure distribution changing, the plain bearing having the most distributed pressure and the lowest, while 3 lobe bearing having the highest pressure (figure 9).
At 305000 rpm the pressure distribution slightly changing (became closer to an equal distribution) for 3 and 4 lobe bearings, while for plain bearing remain the same as at low speed one (figure 10).
Highest pressure is registered at 3 lobe, then 4 lobe and lowest at plain bearing ( figure  11). The pressure levels distribution is same at 105krpm and 305krpm: highest at 3 lobe bearing, lowest at plain bearing ( figure 12 and figure 13).

Conclusions
Cross-coupled coefficients negative value present a tendency to destabilize the rotor ( figure  14 and figure 15). Around 80000 rpm, once the film thickness and eccentricity ratio variation getting close to zero, the pressure distribution stabilizing, keeping the shape until max speed.
While for low speed (rpm) the pressure distribution look very similar between all 3 bearings, at above 100 krpm the 3 and 4 lobe bearings changing the distribution under the geometry influence (lobes). While for 3 lobe bearing the pressure is highest, for 4 lobe the pressure decreasing. For 4 lobe bearing the pressure distribution is more uniform, basically being distributed between lobs.
Between all 3 bearings, the lowest pressure was obtained using cylindrical plain bearing, but a multilobe bearing could match that with a better pressure distribution. All 3 bearings present isotropic behaviour above 40000 rpm, the both damping and both stiffness coefficients being almost equal.
The cross-coupling stiffness coefficients (linearized bearing damping and stiffness coefficients from figure 14) being negative, this shown a tendency to destabilize the rotor system.