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An Application of Newton's Method to the Lubrication
Analysis of Air-Lubricated Bearings
Nenzi Wang a; Chinyuan Chang a
a
Mechanical Engineering Department, Chang Gung University, Tao-Yuan, Taiwan,
R.O.C.

First Published on: 01 April 1999

To cite this Article: Wang, Nenzi and Chang, Chinyuan (1999) 'An Application of
Newton's Method to the Lubrication Analysis of Air-Lubricated Bearings', Tribology
Transactions, 42:2, 419 - 424
To link to this article: DOI: 10.1080/10402009908982237
URL: http://dx.doi.org/10.1080/10402009908982237

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Downloaded By: [Indian Institute of Technology] At: 19:29 18 December 2007

An Application of Newton's Method to the Lubrication

Analysis of Air-Lubricated ~ e a r i n ~ s @
NENZI WANG and CHINYUAN CHANG
Chang Gung University
Mechanical Engineering Department
Tao-Yuan, Taiwan, R.O.C.

This study deals with the development of a computational pro- KEY WORDS
cedure for solving the isothermal compressible Reynolds equation Air Bearings; Gas Bearings; Gas Lubrication
as the governing equation of air-bearing analysis. Newton's
method is used to linearize Reynolds equation and an iterative INTRODUCTION
successive relaxation process is adopted to solve for the air film One of the characteristics of air- or gas-lubricated journal bear-
pressure. The optimal value of relaxationfactor for the cases stud- ings is that the bearings are usually operated at high eccentricities
ied is suggested in this report for numerical stability and compu- under moderate load even with the help of externally pressurized
tational eficiency. The model is verified numerically by examin- (aerostatic) design. To predict the bearing performance under high
ing the conservation of mass flow of the lubricant. The dimen- eccentricities some limiting methods, e.g., pressure perturbation
sional analysis of the governing equation permits the model to be
- -
technique (1)-(3) or linearized ph approximation ( 4 ) , (5), cannot
readily applied to any given film geometry. provide accurate solutions.
In 1991, Tanaka and Muraki (6) studied a high-speed laser
The computer model developed can evaluate the air film pres-
scanner supported by a self-acting aerodynamic bearing. Their
sure distribution, load capacity,frictional force, and mass flow of
experimental findings showed the bearing under testing was run-
an air bearing. The proposed computational scheme eflciently
ning at high eccentricities with low radial roundout. Slocum (7)
analyzes the performance of air-lubricated journal bearings at presented comprehensive design procedures for orifice-compen-
large eccentricity ratios. A similar procedure can be employed to sated gas-lubricated journal bearings based on experimental
investigate the pe$ormance of highspeed noncircular air bearing works. Recently, the effects of the air bearing surface roughness
or gas-lubrication film under slip-flow conditions. This study on the bearing performance were investigated (8), (9). The results
gives an analytical basis for the design of or8ce-compensated confirmed the general belief that surface roughness has insignifi-
externally pressurized air-lubricated bearing. cant effects on the lubrication characteristics of the bearings in the
case of laminar flow. In 1996, Hughes et al. (10) analyzed a gas
lubricated thrust bearing experimentally. Detailed measurements
Presented as a Soclety of Trlboioglsts and Lubrlcatlon Englneers of the flow in an idealized gas thrust bearing were presented. The
paper at the ASMEISTLE Trlbology Conference in
Toronto, Ontarlo, Canada, October 26-28, 1998 pad surface temperature measurements verified that the bearing
Final manuscript approved August 21,1998 flow could be assumed to be locally isothermal.

NOMENCLATURE P. = pressure of externally pressurized air, kg/cm2

6 = modification of F(6" = F"" - F") r =journal radius, m
E = eccentricity ratio R = gas constant of air
P = air viscosity, ~ - s / m ' T = air temperature, "C
P = air density, kg/m3 U = bearing velocity, mls
PO = ambient air density, kg/m3 x = bearing sliding direction
P, = density of externally pressurized air, kg/m3 Y = direction perpendicular to the bearing sliding direction
o = relaxation factor
A = orifice area, m2 Subscripts
A,-& = coefficients in Eq. [lo] 1,m = indices for grid in the x- and y-direction
B = bearing width, m
c, = discharge coefficient of orifice
Superscript
c, = flow correction factor
= minimum air film thickness, p n = index for numerical iteration
h,
L = bearing length (2 x. r), m
m = mass flow rate of air, kg/s Overbars
P = air film pressure, kg/cm2 x,y,p, etc. = nondimensional variables
P. = pressure in the bearing recess, kg/cm2
Downloaded By: [Indian Institute of Technology] At: 19:29 18 December 2007

From the experimental works mentioned above, it is obvious Except for some self-driven units, which require external cool-
th:~t:I computational model, which can accurately predict the per- ing to remove the heat generated in the driving motors, the
formance of high-speed journal bearings at high eccentricities isothermal condition is a reasonable assumption in modeling air-
undcr isothermal condition, is essential for lubrication engineers lubricating bearings (10). In addition, assuming air is an ideal gas,
in dcaling with the design of gas-lubricated bearings. Among the the equation of state of air isp=$. The Reynolds equation for
various numerical approaches, finite difference and finite element isoviscous, compressible fluid can be expressed as follows.
techniques are commonly adopted for lubrication analysis. The
maill advantagc of using finite element method (FEM) for lubri-
cation ~ ~ r o b l e is
~ nitss flexibility in dealing with irregular domains.
For this reason, finite element solutions were widely used in gas- To add to the generality of the governing equation, a procedure
bcaring analysis (11)-(14). However, the formulation and the pro- similar to the study reported by Wang and Seireg (17) was adopt-
gralii coding of FEM are much more complicated than the finite ed to nondimensionalize the governing equation. The following
diffcrcnce method (FDM) with successive relaxation solution parameters were used to nondimensionalize Eq. [I].
sche~iiepresented in this report. Also, the difficulty of FDM in
hnndling complex geometries is overcome by the nondimensional
tnuisformation of the governing equation used in this study. The
tr:oisforrnation permits the computational model to deal with air-
or gas-lubricated bearings with arbitrary film profiles. Substituting Eq. [2] into Eq. [I], yields the following nondi-
In 1985, Gero and Ettles (IS) evaluated FDM and FEM for the mensional form.
steady, isoviscous, incompressible lubrication problem. They pre-
sumcd that for more complicated coupled problems the solution
can bc viewed as solving a sequential series of simple, uncoupled,
and stcady problems. Their results for two-dimensional bearings The pressure boundary conditions of the studied air bearings
showed the solutions calculated by a five-point FDM (scheme are atmospheric pressure. Except under extreme conditions, e.g.,
ilccuracy is second-order in space) have smaller relative errors air-lubricated bearings are modeled at low speeds, Eq. [3] has to
than those obtained by linear four-noded E M . The average CPU be linearized before it can be solved numerically.
lime is 0.15 sec. for five-point FDM and 0.17 sec. for four-noded
IXM. They also concluded that a nine-noded quadratic finite ele- NEWTON'S METHOD AND NUMERICAL ANALYSIS
mcnt provides the best balance between computational cost and Based on the bearing design, the pressure distribution of an air
accuracy. However, a solution scheme with second-order accura- bearing can be obtained when proper boundary conditions are
cy should be adequate for engineering practices when the manu- specified and the governing equation is solved. In this study
fitcturing tolerance of air bearings is considered. Newton's method is implemented to linearize the governing equa-
This study presents a second-order accurate numerical scheme tion, and an iterative successive relaxation process is adopted to
capable of solving finite aerostatic journal bearings with arbitrary solve for the air film pressure. The procedure is described in detail
film profilc operating at high eccentricities. The computer model in this section. Firstly, a nonlinear function in F is defined
clcvcloped can evaluate the air film pressure distribution, load as follows.
capacity, frictional force, and mass flow of air bearings. A similar
proccdure can be employed to investigate the performance of
high-speed noncircular air bearings or gas-lubricating film under
slip flow conditions. From Newton's method, the authors get

GOVERNING EQUATION AND ITS NONDIMENSIONAL

FORM
Thc mathematical model of air-lubricated bearings as com- where 6" =pn"-8" . Now, expand / ( ~ + / 3 6 ) in a Taylor series
pared with fluid-film bearings is much more difficult to handle about p , i.e.,
duc to the con~pressibilityof the air. The generalized Reynolds
ccluntion (16) for compressible lubricant is simplified and used to
~iiodelair-lubricated bearings in this study. When the variation of
viscosity and density across the film direction is neglected, the Next, by substituting p + p 6 into Eq. [4], another expression
cquation is reduced to a nonlinear elliptical partial differential f o r f ( p + p ) can be written as
cquation as shown below.
Downloaded By: [Indian Institute of Technology] At: 19:29 18 December 2007

An Application of Newton's Method to the Lubrication Analysis of Air-Lubricated Bearings

From Eq. [5], calculating the first derivative of r(p+~s) with

respect to p and setting p = 0, yields

From Eq. [6], the first derivative of f(p+Ps) with respect to

p is

Be letting P = 0, the authors have

From the right-hand sides of Eqs. [7] and [8], the original non-
linear problem in F is reduced to solving a linear partial differen-
tial equation in 6 (Eq. [9]) in order to determine the value that
improves the accuracy of the estimate 6 The initial values used in the computational model are
&=a , and $ m = ~ . ~ . The successive relaxation method is
. applied to solve 6,., in Eq. [lo]. The formula can be written as
$ ( p ~ ~ $ + ~ ~ $ ) + 5 ( . ~ $ + & ~ $ ] -a(@)
6 ~
-(AIJ?G!. + A,~LI,. + A,~,%:I + lh6(,-1 +A+5[11 - 4.) [I 11
a --,+ a --,a?
= - x [ ~ h $]-%[ph %]+6?
a(S) [91 g, =s;;l +o,
5

Equation [9] can then be solved by the successive relaxation where w is the relaxation factor and the superscripts of 6 denote
method. The method chosen is based on its easy programming and the indices for numerical iteration. The air film pressure is updat-
fast convergent rate. Since 6 is a function of p , an iterative pro- ed by using PI? = +Qm . 'The process is repeated until the
cedure is required to have both 6 and p converged. The partial pressure is converged or 6h approaches zero. The overall
differential terms in Eq. [9] are replaced by second-order accurate scheme accuracy is second-order in space.
finite difference formulas as expressed below. The relaxation factor used in this study has decisive impor-
tance in solving Eq. 11 I]. The choice of an optimum value of w is
a rather complex task. In this study, the relaxation factor is deter-
mined by numerical experimentation. It can be seen later in this
report that underrelaxation, for <MI, is used to make a noncon-
vergent iterative process converge.
The conservation of the mass flow was monitored in the com-
puter model to ensure the accuracy of the numerical methods and
the chosen grid sizes. The mass flow equations used are

The resulting system of equations is shown in Eq. [lo]

where m, is the mass flow in the sliding direction, and my is the
mass flow in the direction perpendicular to the sliding direction.
The formulas were also used to calculate the flow out of the
recesses in the externally pressurized gas-bearing analysis.
Downloaded By: [Indian Institute of Technology] At: 19:29 18 December 2007

Bearing Diam. = 3.81 a

Ndth = 3.81 a
Clearance = 9.5 r m

Circumferential Location, deg.

Flg. 2--Calculated pressure dlstributlons of journal bearing with dlffer-

ent computational models.

Flg. 1-Externally pressurlzed three-recess air-lubricated journal bear- Vmin (8.72 x 10-7 kglmin) and the out flow is 7.07 x 10-4 Vmin
ing. (8.49 x 10-7 kglmin). The mass flows are small due to the small
NUMERICAL SIMULATION bearing clearance.
The studied orifice-compensated aerostatic bearing has diame- The computer program was written in Fortran 90 and run on a
ter 38.12 mm, bearing width 38.12 mm with an equally spaced Pentium PROl200 IBM compatible PC. Several relaxation factors
three-recess design as shown in Fig.1. The width of each recess is were tested. The results are plotted in Fig. 3. It can be seen that the
cqual to the bearing surface between two recesses. The pump sup- solution converges faster with a larger relaxation factor. However,
plied air pressure (p,) is 4.14 kg/cm2.By assuming air temperature the solution diverges as the relaxation factor exceeds 0.75. The
is 2 1 "C and atmospheric pressure is 1.01 kg/cmz,the density of the value of relaxation factor used in all other cases is 0.5 for numer-
air is 0.001201 kg11 and the viscosity is 0.0000179 ~ - s / m ' .The ical stability and computational efficiency. Figure 4 shows the
grid sizes used were 91 in the x-direction and 41 in the y-direction. peak pressure of aerodynamic journal bearing under various
The recess pressure is calculated as follows eccentricity ratios. The pressure increases exponentially as the
eccentricity changed from 0.3 to 0.95. The conservation of mass
flow of lubricant is verified. This demonstrates the studied com-
putational procedure using Newton's method can model bearings
with high eccentricities. Figure 5 shows the performance of the
whcrc the flow correction factor, C, = 0.98, discharge coefficient, studied air lubricated journal bearing at speeds of up to 20,000
C, = 0.62, and the orifice diameter used in the analysis was 2 mm. rpm. Under high-speed operations, the peak pressure as well as
The size of the orifice is designed to maintain the pressure ratio the load carrying capacity of the aerodynamic bearing reaches
(recess pressure to the pump supplied pressure) above the critical asymptotic values. This confirms the results found in the literature
prcssurc ratio, which is 0.528 for the air. In this arrangement, the that the canying load becomes independent of speed at extremely
local sonic flow is eliminated. In the analysis, the recess pressure high speeds.
is obtained through an iterative procedure. At the beginning of the Figure 6(a) shows the aerodynamic effect is negligible at 30
iterative process, assunic that the recess pressure is the same as the rpm in an aerostatic bearing, and the recess pressure of the two
pump supplied pressure. Based on this recess pressure, the pres- recesses away from the minimum film thickness position is
sure distribution of the air film is calculated according to the reduced from pump supplied pressure due to the larger clearance.
numerical scheme described in the previous section. Once the The reduced pressure in the recesses is calculated by using the
pressure distribution is Found, the mass flow out of the recess can iterative procedure described in the beginning of this section. The
be calculated using Eqs. [I21 and [13]. Then, the new recess pres- selection of the orifice size is to avoid the occurrence of local
sure is estimated by Eq. 1141. This changes the boundary condi- sonic speed. If the orifice is too small the flow will reach sonic
tions of the air film. The process is repeated until the recess pres- speed due to small pressure ratio. For the case of the bearing run-
sure is converged. ning at 30 rpm, the in flow is 2.748 Vmin (3.30 x 10-3 kglmin) and
When the compressibility of the air is not included in the the out flow is 2.815 Vmin (3.38 x 10-3 kglmin). Figure 6(b) is a
model, for the lubricant with the same viscosity as the air, the polar plot of pressure in an aerostatic bearing running at 3000
incompressible model predicts much higher peak pressure as rpm. Figures 6(c) and 6(d) show the pressure distributions of the
shown in Fig. 2. Cavitation is not present in an air-lubricated bear- three-recess aerostatic journal bearing running at 30 and 3000
ing as clearly shown in Fig. 2. When the air bearing is running at rpm, respectively. The aerodynamic effects can be seen from Fig.
E = 0.9 and 10,000 rpm, the flow into the bearing is 7.26 x 10-4 6(d) when the bearing speed is 3000 rpm.
Downloaded By: [Indian Institute of Technology] At: 19:29 18 December 2007

An Application of Newton's Method to the Lubrication Analysis of Air-Lubricated Bearings 423

ai
E -

I I

-----L-----L-----'-----,-----l.-I---J-----J------'------'----
20 , , ! , , t I 4 (
, , , , I I I I I
, , , , I I I t I
, , , , I , I I I
, , , , O I I D
, , , , I I I I I
1 . I ,
0 ' I ' I ' l ' i ' I B I . i L
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Relaxation Factor Eccentricity Ratio
Flg. 3-Computation time of successive relaxation method for various Fig. &-Peak pressure at various eccentricity ratlos of an alr-lubricated
relaxation factors.
journal bearing.

I I I I

- Bearing Diam. = 3.81 an

Wdth = 3.81 an
- Clearance = 9.5 pm
. Eccenlricity = 0.9

0 1 I I I
' 10
0 5.000 10,000 15,000 20,030
Bearing Speed, rpm

Fig. 5--Peak pressure and load carrying capacity vs. bearing speed.

CONCLUSIONS extra load support and maintain accurate shaft position at low
By careful selection of the relaxation factor, the proposed speeds or under standstill condition. However, the instability
numerical procedure can efficiently solve the Reynolds equation caused by pneumatic hammer in high-speed recessed air bearings
for isoviscous, compressible lubricant. The solution scheme used and the effect of 3-D air flow in the recesses on the bearing per-
is second-order accurate in space. The successive relaxation formance deserves further study.
method used in this study is easy to program as compared with the
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45 3,000 rpm /\
1 -/ 4.0

3 - 3.5

3.0 :.,,,------------
30 rpm
.---
,' -- - _ _ - _ _ _ _ _ _ _ _ _ _ ,
-.-
g 2.5 1
a
E
- 2.0 - Bearing Diam. = 3.81 cm
1.5 - Mdlh = 3.81 cm
Clearance = 9.5 prn
1.0 Earentricky = 0.9
No. of Remssas = 3
0.5 -
0.0
0 90 160 270 360
Circumferential Location, deg.

Fig. 6(a)-Calculated pressure distributions along the centerllne of the

aerostatic bearing in the direction of sliding.

Pressure Flg. 6(c)-Pressure dlstrlbution of aerostatic journal bearing (30 rpm, E

= 0).

Fig. 6(b)-Caicuiated pressure distribution of aerostatic bearlng (3000

rpm, E = 0.9).

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