Gas Lubricated Journal

7/28/2019 Gas Lubricated Journal

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Chapter 15Gas-Lubricated JournalBearingsSymbols

//'

i n t e g r a t i o n constantsbearing number f o r foil bearings,

Af

width of bearing, dimension inside-leakage direction, mgroove width, mr a d i a l clearance, mpivot c i r c l e clearance, mdimensionless friction force,/Vp.cfriction force,Nfriction force p e r unit width,N/mdimensionless f i l m thicknessHim thickness ratio, ^s/^trf i l m thickness, mc e n t r a l f i l m thickness, mminimum f i l m thickness, mo u t l e t f i l m thickness, mr a t e of working against viscousshear (power toss), Wfi l m thickness in r i d g e region, mf i l m thickness in step or grooveregion, mlength in 2 direction, mdimensionless stability parameter

ma mass of body, kgm^ mass of body per u n i t width,kg/mA^o number of pads or groovesP dimensionless pressurep pressure, PaPa ambient pressure, PaP1,P2 Hrst-, second-,..., order pertur-

bation pressure, PaR radius, mR gas constantr radius, mT tension in f o i l , Nt^ temperature, Ct q torque, N-mu v e l o c i t y , m/sM b surface & velocity, m/sPF dimensionless loadWr dimensionless resultant load of a

journal bearingW^ dimensionless resultant load in a

journal bearing when side l e a k -a g e i s neglected

Wr dimensionless r e s u l t a n t first-order-perturbation loadWx dimensionless t a n g e n t i a l load

W; dimensionless normal load

369

Copyright 2004 Marcel Dekker, Inc.


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? ?7 / 0

r e s u l t a n t load in a journalbearing, Nr e s u l t a n t load p e r unit lengthin a journal bearing, N/mt a n g e n t i a l load, Nt a n g e n t i a l load p e r unit l e n g t h ,N/mnormal load, Nnormal load per unit length,N/mCartesian coordinate i n d ire c -t i o n of s l i d i n g , mCartesian coordinate i n d ire c -t i o n of side leakage, mCartesian coordinate in direc-t i o n of Him, mgroove width ratio, s/( - )angular extent of pivoted pad,deggroove angle, deggroove width ratio, & i / be c c e n t r i c i t y ratio, e/ccoordinate defined in Eq. (15.1)absolute v i sc o si ty , P a - sabsolute viscosity at p =0 andconstant temperature, P a - s

coordinate i n c y l i n d r i c a l polarcoordinatesfunction of Aj defined in Eq.(15.25)f u n c t i o n of Aj defined in Eq.(15.26)dimensionless bearing numberfor journal bearingswidth-to-diameter r a t i oc o e ff i c ie n t o f s l i d i n g frictionangle between l i n e of centers,degd e n s i t y , kg/mshear stress, Paattitude angle, degangle i n spherical polar coordi-nates, degangular extent from i n l e t top i v o t location, degangular v e l o c i t y , rad/s

Subscriptsso l i d a (upper surface)so l i d & (lower surface)r i d g estep

15.1 IntroductionIn Chapter 14 the focus was on the fundamentals of gas lubricationin generaland on applying these fundamentals to thrust bearings. This chapter contin-u e s with gas-lubricated bearings, b u t t h e concern here i s with journal bearings.Recall that journal bearing surfaces a r e p a r a l l e l t o t h e axis o f rotation, hereast h r u s t bearing surfaces a r e perpendicular t o that a xis. S e l f - a c t ing journal bear-ings, considered i n this chapter, re l y o n shaft motion to generate th e load-supporting pressures i n t h e l u b r i c a n t fi l m. General information about journalbearing operation i s given a t t h e beginning o f Chapter 1 0 a n d therefore w i l l n o tbe repeated here.

15.2 Reynolds EquationAssuming no s lip at the f l u i d- su rfa c e interface, isothermal conditions, and time-i n v a r i a n t lubrication, t h e compressible Reynolds equation i n Cartesian c o o r d i -

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nates can be written from Eq. (7.55) as

For journal bearings, as considered in Chapter 10(see Fig. 10.2), it is convenientt o change the variablesin the preceding equation to

2=r^ y =r( M& = ruj (15.1)Substituting Eq. (15.1) into Eq. (7.55) gives

d

I f ppaP and A =c#t h i s equation becomes

where

i s the dimensionless bearing number for ournal bearings.

15.3 Limiting SolutionsJust as for gas-lubricated thrust bearings in Chapter 14, two l i m i t i n g cases areconsidered here.

15.3.1 Low Bearing NumbersAs in Chapter 14, as the speed a? 0 and A., 0, P 1 and the pressurerise AP 0.

Therefore, expanding Eq. (15.3) and neglecting these terms results in

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This i s t h e same e q u a t i o n d e a l t w i t h i n Chapter 1 0 f o r incompressibly u b r i c a t e dj o u r na l b e a r i n gs . N e g l e c t i n g the s i d e - l e a k a g e term in Eq. (15.5) and assuminga full Sommerfeld s o l u t i o n from Eq. (10.30) g i v e s

From the friction force the friction coefficient isc ( l2e2)MA,-o- (15.7)

These r e s u l t s a r e a p p l i c a b l e f o r L < j 0 o r A., 0 f o r a self-acting, g a s l u b r i c a t e dj o u r n a l b e a r in g .15.3.2 High Bearing NumbersA s i n Chapter 1 4 , a s oj oo o r Aj oo, t h e o nl y w a y f o r t h e p r es su r e toremain finite in Eq . (15.3) is for

i m p l y i n g t h a t J**R = cons tant. A p o s s i b l e s o l u t i o n i s .P t x 1/R, and .P iss y m m e t r i c a l about t h e line o f c e n te rs . I n Chapter 1 0 t h e film t h i c k n e s s wagd e r i v e d a s

) (10.5)#=-=!+ cos < ^ (15.9).'.F=- (15.10)1+ OS 0, # constant = A

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(x ^ (15.12)From Eqs. (15.9) and (15.10) for a concentric journal (e = 0)

P=R= =1 (15.13)Therefore, Eq. (15.10) g i ve s

P=- (15.14)The normal load-carrying capacity for an infinitely wide journal bearing whenA' oo i s

/ COSWo rn+COSUsing the Sommerfeld substitution covered in Chapter 10 [Eqs. (10.9), (10.11),and (10.14)] gives

The friction f o r c e for an infinitely wide journal bearing when oo or A., > ooi s/-=-r/J o

In dimensionless form t hi s equation becomes

But 2? r^ G%( f^0 7o ^

The friction coefficient when ^ oo and A., oo is

,_ 3(1-

3 [1_ (1 ,2)1/2] .These l i m i t i n g s o l u t i o n s are important information when verifying exact nu-merical solutions.

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15.4 Pressure Perturbation SolutionAusman (1959) used the perturbation method to l inearize Eq. (15.2) to obtainan approximate s o l u t i o n . The genera,! concept of the perturbation method is tos u b s t i t u t e Eq. (10.5) and

P = p a + pi+e2p2 + ... (15.18)into E q . (15.2) an d neglect a l l terms of order e^ o r higher. This gives

(15.19)Equation (15.19) can be solved (see Ausman, 1959) for the first-order pertur-bation pressure, which i n turn c a n b e i n t e g r a t e d i n th e u s u a l fashion t o obtainpara l l e l and perpendicular components of the load tD x and il^. The results are

= _=_JA+ /,(A,,A,)1 (15.20)-[A-/;(A.,,A.,-)] (15.21)

whereA., = (15.22)

^ , - A \ \ (Ah-AeA.,)sin(2AcA.,)-(AbA,,+Ac)sinh(2AhA,,) ^-^r,\/ (A.,,A.,j \ / i , A2\l/2 [ t , r o A \1 I ' g A \ . - j l (lo. jj, _ (Ah - AcA^) sinh(2AhA.,)- (A^A^+Ac) sin(2AeA^) / i t - ^ \

" ^' ** A,,(l +A2)V2 [cosh(2A^A,,)cos(2A,A^)] ^ ' ^

(15.26)The equations for the tota l l o a d and the a t t i t u d e angle can be written as

IV$=tan^-^ (15.28)WNote from Eqs. (15.20) and (15.21) that the A r s t - o r d e r perturbation s o l u t i o n

y i e l d s a l o a d l i n e ar l y r e l a t e d t o t h e e c c e ntri c i t y ratio e. This i s a consequenceo f the l i n eari zation and i s v a l i d only for small e, say e < 0.3, although as aconservative engineering approximation i t m a y b e used f o r higher values.

Figure 15.1 plots th e l o a d parameter a n d t h e attitude angle f o r t h e first-order perturbation a n al y s i s a t several Aj values.



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100

80

60

40

20

Attitude a n g l eLoad

0 .1 .5 .7 1 1 .5 2D i m e n s i o n l e s s b earing number, A y

5 10

Figure 1S.1: Design chart fo r r a di a l l y loaded, se lf-a cting, gas-lubricated j o u r n a lbearings (isothermal first-order perturbation solution.) [.R"cm Amman

Example 15.1Given A gas l u b r i c a t e d journal bearing with the Reynolds equation in theform of Eq. (15.3).Find Derive the Reynolds equation for a first-order perturbation oneccentric-ity when

$PR^=1+g^i and R=1+e cos (a)Solution

Also we can write thatPR =

9(PR)_ 1d< ^ *R dR

Substituting Eqs. (a) through (d) into Eq. (15.3)gives- + = iR3 d^yj R3( 9(y* R R2



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or

Expanding gives

+ T7 I 1- ) -

Multiplying by R^ gi v es

Now we want toperturb ^so that

Also

Note t h a t ()',S,()',and ()'are of o r d e r e' and h i g h e r andt h u s are e l i m i n a t e d . Making use of Eq s. (/)through (4;), hile n e g l e c t i n g termsof e^ and h i g h e r , Eq. (e) becomes

( 1e$i)(l + cos^ei +2(1 +^+



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Neglecting terms of order e^ and higher gives

In t h i s equation, ^i is the desired value, from which we can calculate ^from

Also

15.5 Linearized ^ SolutionAusman (1961) introduced a linearized p/t solution to the self-acting, gas-l u b r i c a t e d journal bearing problem that corrected for the deficiency of the A r s t -order perturbation s o l u t i o n a t h i g h - e c c e n t r i c i t y ratios given i n t h e last section.The general method of linearization is essentially the same as the perturbationmethod except that the product p/t is considered to be the dependent v a r i a b l e .To do this, Eq. (15.2) isarranged so that p always appears m u l t i p l i e d by / t .Equation (10.5) and

p =paC+A(p^) (15.29)a r e then substituted into t hi s equation, and only first-order terms in eccentricityratio 6 a r e retained. I t i s assumed that A(p/t) i s o f order e. T h e r e s u l t i n g" l i n eari z e d p/!," equation i s

+ T=A,-- -ep.ccos^ (15.30)Equation (15.30) is es s entia l l y the same form as the first-order pressure pertur-bation equation [Eq. (15.19)] and can be solved in the same manner for p/t.



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Once p/ 3 is f o u n d , the p r e s s u r e can be o b t a i n e d by d i v i d i n g by . The r e s u l t i n ge xpression for p is t h e n put into the l o a d component i n t e g r a l s to o b t a i n Wg andW z - The r e s u l t s are

-(l -


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1.4.

1. 2

1.0

O E x p e r i m e n t a l d a t a (fromS t e r n l i c h t a n d E l w e l l . 1958)*? j Computer s o l utions

Figure 15.2: Effect of di-mensionless l o a d on ec-centricity ratio for fm ite-F i r s t - o r d e r p e r t u r b a t i o n length, self-acting, gas-tubricated journal bear-ing. Dimensionless bear-ing number Aj , 1.3;

i I I width-to-diameter ratio.4 .6

Eccentricity r a t i o , e 1 . 0 A.,, 1.5.

2.5

From Fig. 15.1 for A^ =5 and A., =1=0.7 or =0.7-p^&(2r)e =68.? N

= 35Also, e i s obtained a s

From Eq. (15.33),

r 137.4[1-(l-0.0.80(1 - 0.64)^/2 [ 1 - (0.64)(0.33)]i/292 N



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15.6 Nonplain Journal BearingsThus far this chapter has been concerned with plain gas-lubricated ournal bear-ings. A problem with using these bearings is their poor stability ch aracteristics .L i g h t l y loaded bearings that operate a t low ec centricity ratios a r e subjected tofractional frequency w h i r l , which can r e s u l t in bearing destruction. Two typesof nonplain gas-lubricated journal bearings that find widespread u s e a r e t h epivoted pad and the herringbone groove.

15.6.1 Pivoted-Pad Journal BearingsPivoted-pad bearings were first introduced in Sec. 9.4 as a type of incompress-i b l y l u b r i c a t e d thrust bearing. Pivoted-pad journal bearings a r e most frequentlyused a s shaft supports in gas-lubricated machinery because o f their excellentstabi lity characteristics. A n i n d i v i d u a l pivoted-pad assembly i s shown i n Fig.15.3. A three-pad pivoted-pad bearing assembly is shown in Fig. 15.4. Gener-a l l y , each pad provides pad r o t a t i o n a l degrees of freedom about three orthogonalaxes (pitch, ro l l , and yaw). Pivoted-pad bearings are complex because of themany geometric variablesi n v o l v e d in t he i r design. Some of these variables are

1. Number of pads No2. Angular extent of pad ap3 . Aspect ratio of pad r/&4. Pivot l o c a t i o n


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Pivot

Figure 15.3: Geometry of i n d i v i d u a l pivoted-pad bearing. [From Gunner

F i e l d maps must be generated for other geometries and bearing numbers. Ad-d i t i o n a l field maps are given in Gunter et al . (1964).

When the i n d i v i d u a l pad c h a ra c t eristi c s are known, the characteristics oft h e multipad bearing shown in Fig. 15.4can be determined. With the arrange-ment shown in Fig. 15.4 the load is directedbetween the two lower pivoted pads.F o r this case t h e l o a d carried b y each o f t h e lower pads is initial ly assumed tobe Mj,-cos/?p. The pivot film thicknesses /tpj and /tp^ are obtained from Fig.15.5b. Furthermore, the upper-pad p i v o t film thickness 3,the e c c e ntri c i t yratio e , and the dimensionless load W,-3 can be determined.

Pivoted-pad journal bearings a r e usually assembled with a pivot c i r c l e c l e a r -ance c' somewhat less than the machined-in clearance c. When c'/c < 1, thebearing i s said t o b e "preloaded." Preload i s usually given i n terms o f a preloadcoefficient, which is equal t o (c e')/c. Preloading is used to increase bearingstiffness and to prevent complete unloading of one or more pads. The latterc o n d i t i o n can lead to pad flutter and possible contact of the pad l e a d i n g edgea n d t h e shaft, hich i n turn c a n resu lt i n bearing failure.



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- Pivot

- Pivoted p a r t i a ljournal bearing

F i g u r e 15.4: Geometry o f p i v o t e d - p a d j o u r n a l b e a r i ng w i t h t h r e e pads.^ a / .

15.6.2 Herringbone-Groove Journal BearingsA A x e d - g e o m e t r y b e a r i n g t h a t has d e m o n s t r a t e d good stability characteristicsan d t h u s p r o m i s e f o r u s e i n h i g h - s p e e d gas b e a r i n g s i s t h e h e r r i n g b o n e b e a ri n g.I t consists of a c i r c u l a r j o u rn a l a n d b e a r i n g s l e ev e w i t h s h a l l o w , h e r r i n g b o ne -shaped groov es c u t into either member. F i g u r e 15.6 shows a partia l l y g r o o v e dh e r r i n g b o n e j o u r n a l b e a r i n g. Th e g r o ov e p a r a m e t e r s u s e d t o d e fi n e this b e a r i n g

1 . Groove a n g l e /?^2 . Groove width ratio a&=^3 . Film t h i c k n e s s ratio R =4. Number of groov es Nn5 . Groove width r a t i o = &i

f,.^



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Eccentricity r a t i o , e

A n g l e betweenl i n e of c entersand padl eading e d g e , C o n v e r g i n g -divergingf l u i d f i l m/ ./ / C o n v e r g i n g, f l u i d f i l m

^-Optimumpivot_ l o cation( a ) ! I 1 * f I ! I

.58 .60 .62 .64 .66 .68 .70 .72 .74 .76D i m e n s i o n l e s s pivot l oc ation,

100I I [ I I I I ! I

.58 .60 .62 .64 .66 .68 .70 .72 .74 .76D i m e n s i o n l e s s pivot l ocation, < t ) , / o t p

F i g u r e 15.5: Charts for d e t e r m i n i n g l o a d coefficient, p i v o t Him thi c kness, andt r a i l i n g - e d g e Him thickness. Bearing r a d i u s - t o - l e n g t h r a ti o r/&, 0.6061; a n g u l a re x t e n t of pad ap, 95.5; dimensionless b e a r i n g number Aj, 3.5. (a) Dimension-less l o a d ; (b) dimensionless p i v o t Him thi c kness. [From Cimier e^ a^



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.9 -

.8

.7

b

0.60i y.70

F i g u r e 15.5: on-c l u d e d , (c) Dimen-

.58 .60 .62 .64 .66 .68 .70 .72 .74 .76 sionless outlet film( c ) I I ID i m e n s i o n l e s s pivot location, < t < p / o t p thickness.

F i g u r e 15.6: C o n f i g u r a t i o n of c o n c e n t r i c h e r r i n g b o n e - g r o o v e j o u r n a l b e a r in g .

T h e o p e r a t i n g p a r a m e t e r s u s e d f o r h e r r in g b o n e j o u r n al b e a r in g s a r e1. W i d t h - t o - d i a m e t e r ratio A , , =b / 2 r .2 . D i m e n s i o n l e s s b e a r i n g number Aj =6%^r^/Pa^r-

Both t h es e p a r a m e t e r s a r e d i m e n si o n l e s s .F i g u r e 15.7 presents t h e optimum h e r r in g b o n e j o u r na l b e a r i n g g r o o v e p a -

r a m e t e r s for maximum radial l o a d . These results were o b t a i n e d from Hamrocka n d Fleming (1971). T h e t o p p o r t i o n o f e a c h p a r t i s f o r t h e g r o o v e d memberrotating, a n d t h e bottom p o r t i o n i s f o r t h e smooth member rotating. T h e o n l yg r o o v e parameter no t r e p r e se n t e d in this figure i s t h e number of g r o o v e s t o b eused. Hamrock and Fleming (1971) f o u nd t h a t the mwwwM?n number of g r o o v e st o he p l a c e d around the j o u rna l can be represented by No > A.,/15.F i g u r e 15.8 e s t a b l i s h e s th e maximum normal l o a d - c a r r y i n g c a p a c i t y f o rt h e o p e r a t i n g p a r a m e t e r s o f a g a s - l u b r i c a t ed h e r r i n gb o n e j o u rn a l b e a r in g . T h eoptimum g r o o v e p a r a m e t e r s o b t a i ne d from Fig. 15.7 are assumed.



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Wi d t h- t o -d i a m e t e r ratio,

40 80 120 160 0 40D i m e n s i o n l e s s b e a r i n g number, A . 120 160

Figure 15.7: Charts for determining optimal herringbone journal bearing grooveparameters formaximum ra dia l load. Top plots are forgrooved member rotat-ing; bottom plots are forsmooth member r o t a t i n g , (a)Optimal Rim thicknessratio; (b) optimal groove width ratio. [.From FamrocA; and

More than any other factors, se lf-excited whirl instabi lity and low loadcarrying capacity l i m i t the usefulness of gas-lubricated journal bearings. Thew h i r l problem i s t h e tendency o f t h e journal center t o o r b i t t h e bearing centerat an angular speed less than or equal to one-half that of the journal abouti t s own center. In many cases the whirl amplitude is l a r g e enough to caused e s t r u c t i v e contact o f t h e bearing surfaces.

Figure 15.9, obtained from Fleming and Hamrock (1974), shows the stabil-ity attained b y t h e optimized herringbone journal bearing. I n this figure t h edimensionless stabi lity parameter i s introduced, here

(15.37)

where m^ is the mass supported by the bearing. In Fig. 15.9 the bearings withthe grooved member rotating aresubstantially more s t a b l e than those with the



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1.0

.8

.6

'".2

Wi dth -to -r a t i o .

- 1/ 4 26^ ^. -, -\ 1A \ 2 24\ _^_ ** ' .;/-*** S TY" ^

L ^ ^ 1 ' #1,

f\r\Y /*! ./*i.

i t i i

< " ""120 160 0 40D i m e n s i o n l e s s b e a r i n g n u m b e r , L

Figure 15.7: Concluded, (c) Optimal g r o o v e l e n g t h ratio; (d) o p t im a l g r oo v eang le .

smooth member r o t at i n g, e s pe c i al l y a t h i g h b e a r i ng numbers.

15.7 Foil BearingsA u n i q u e b e a r in g f o r h i g h - s p e ed a p p l i c a t i on s i s t h e f o i l b e a r i n g , shown schemat-ically in Fig. 15.10a. Foil bearings have found u s e i n a e r o s p a c e a p p l i c a t i o n s fortheir light w e i g h t , a n d have dire c t a p p l i c a t i on t o t h e d e s i g n o f t a p e r e c o r d i ngheads. The main advantages o f f o i l bearings ar e their self-acting n a t u r e , theirlight w e i g h t and their ability t o u s e a i r a s t h e l u b r i c a n t for light ly loaded app l i-cations. Some o f t h e concerns fo r f o i l b e a r i n g s a r e t h e tendency f or w h i r l andrelatively tow l o a d s u p p o r t . A number o f investigators have modeled f o i l b e a r -i ng s , n o t a b l y Blok and van Rossum (1953), Walowit and Anno (1975), Gross,et a l . (1980), a nd Bhushan (2002). Th e main differentiating eature o f a f o i lb e a r i n g i s t h a t o n e o f t h e b e a r i n g s u rf ac e s i s extremely c o m p l i a n t , c o n s i s t i n gof a t h i n fi l m o f metal o r polymer. A s c a n be seen , t h e f o i l b e a r i n g c o n t a i n sa n inlet region where t h e film shape i s onverging , a centra l region where t h efilm t h i c k n es s i s e ss en ti al l y c o n s t a nt , a n d a n exit r e gi o n. T h e c e n t r al r e g io ni s of p a r t i c u l a r i nt er e st , s in c e smooth p ara l l e l surfaces are r a r e l y encountered



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.. -

1 2

130m^ 16

12

Figure 15.8: Chart for determiningmaximum normal load-carryingca-pacity. (a) Grooved member rotat-

40 so 120 160 ing; (b) smooth member rotating.Dimer-sionless b e a r i n g n u m b e r , and

in bearings because o f t h e lack of load support, as discussed in Sec. 8.1. Also,t h e performance of magnetic tape recording heads depends on the abi lity ofmaintaining a constant and uniform film thickness.

The usual forms of a f o i l bearing are or either Ma =0 (for a ournal bearing)o r M b = 0 (as commonly seen in magnetic heads). Consider the case where= 0 and t t a = M , with air serving as the lubricant. It w i l l be assumed that

th e pressures are low, so that air can be taken as isoviscous and incompressible.The f o i l or tape has a constant tension, T, and wraps pa r t i a l l y around thebearing. Considering an element of the bearing as shown in Figure 15.10b,e q u i l i b r i u m y i e l d s (15.38)For small 0, s i n # 0so that Eq. (15.38) becomes

pR =T; p=I (15.39)This pressure w i l l b e constant throughout t h e entire central region o f t h e f o i lbearing. Assuming no s i de How, the Reynolds equation in the inlet zone of thef o i l bearing is



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20

1 3

. 1.08.06

.04

.02

U n s t a b l e

^A \Smooth member rotating - - \ \\*\

Wi d t h - t o -d i a m e t e rr a t i o ,/41/212

4 6 8 10 20D i m e n s i o n l e s s bearing n u m b e r , A^

40 60 80

Figure 15.9: Chart for determining maximum s t a b i l i t y of herringbone-groovebearings. [From Hemmy an^ #amroc%:



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DeAne the bearing c h a ra c t eristi c number for a f o i l bearing as

So that Eq. (15.47) becomes

The boundary conditions are that p =E/T at the edge of contact [from Eq.(15.39)] and p = 0 far from the contact region. These can be written in termsof ^and ( as

a t ^=0, ^ =at $=oc, < ? ! < =0

Therefore, integratingEq. (15.49),

The integra l on the righ t si de of Eq. (15.50) can be shown to be equal to Tr/16.Therefore, applying the boundary conditions,

R/ = (15.51)S u b s t i t u t i n g Eq. (15.48) into (15.51) and s o l v i n g for the centra! Aim thicknessy i e l d s

" 16 , ,7rV2

S o l v i n g f o r A c ,,.,,(15.53)

T h e Him thickness decreases s lightly i n t h e o u t l e t region i n order t o maintainflow continuity. Gross et al (1980) have derived the pressure distribution acrossa f o i l bearing, and their resu l t s are shown in Figure 15.11. Bhushan (2002)r e p o r t s t h e minimum fi l m thickness i n t h e f o i l bearing a s

/tniin 0.72/tc (15.54)



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a

Pres s ure

F i l m t h i c k n e s sI I ! I I I I I I I I I I !!! I I t I

1.0 .0.8 m0.6 %0.4 n0.2 o

oJi 4 0 2 4 6

I n l e tregion D i m e n s i o n l e s s position,64024

E x i tregion

Figure 15.11: Pressure distributiona n d film thickness i n a foil bearing.

15.8 ClosureThe discussion of g a s - l u b r i c a t e d bearings was continued from Chapter 14 witht h e focus of this chapter being journal bearings. The appropriate Reynoldsequation was presented, and l i m i t i n g s o l u t i o n s of extremely low- and high-speedr e s u l t s were discussed. Because of the n o n l i n e a r i t y of the Reynolds equation,two approximate methods were introduced. The f i r s t of these approaches is aperturbation s o l u t i o n where t h e pressure i s perturbed with t h e e c c e ntri c i t y ratioe . When terms of order e^ and higher are neglected, the nonlinear Reynoldsequation becomes l i n e ar , and thus a n a l y t i c a l solutions can be obtained. It wasfound that t h e first-order perturbation s o l u t i o n yieldsa load r e l a t i o n s h i p that i sl inear l y r e l a t e d t o t h e eccentricity ratio. This r e s u l t w a s found t o b e v a l i d onlyfor small eccentricity ratios, say e < 0.3, although as a conservative engineeringapproximation i t m a y b e used f o r higher v a l u e s .

The second approximate method covered in this chapter was the linearizedp/: approach of Ausman. The l i n eari zation isessential l y the same as the per-t u r b a t i o n method except that the product pA is considered as the dependentv a r i a b l e . This method predicts the load-carrying capacity more accurately fort h e complete range o f e c c e ntri c i t y ratios. Furthermore, t h e results agree wellwith those found experimentally. The l atter part of the chapter covered non-p l a i n journal bearings. Two types of nonplain g a s - l u b r i c a t e d journal bearingwere considered, namely, pivoted pad and herringbone groove. Charts werepresented t o h e l p i n t h e design of these bearings. Foil bearings were analyzed,and Rim thickness equations used in the design of magnetic recording systemswere developed.

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15.9 Problems1 5.1 D i s c u s s t h e effect o f s i d e l e ak a ge o n normal l o a d - c a r r y in g c a p a c i ty o f g as -

and l i q u i d - l u b r i c a t e d j o u r n a l b e a ri n g s.15 . 2 S t a rt i ng w i t h Eq. (15.2) show a l l t he s te ps i n a r r i v i n g a t t h e " linearized

pA" s o l u t i o n g i v e n in Eq. (15.30).15.3 Starting with Eq. (15.3) deri v e the Reynolds equation for a perturbed

solution.15.4 A s e l f -a c t in g , a i r - l u b r i c a t e d journal b e a r i n g with a 400-mm d i a m e t e r and100-mm w i d t h is u s e d in gas t u r b i n e aircraft engines. It may he assumed

t h a t an e c c e n t r i c i t y ratio o f 0 . 5 i s i ts o p e r a t i n g p o s i t i o n . Estimate theth eoreti c a l maximum l o a d c a p a c i ty of the b e a r i n g :(a) Under f u l l - s p e e d c o n d i t i o n s at sea l e ve l (engine speed 20,000 rev/min;

ambient pressure 101.3(b) Under e n g i n e i d l i n g c o n d i t i o n s at 13-km a l t i t u d e (speed 10,000 rev/min:

pressure 16. 5 815.5 A magnetic r e c o r d i n g head u s e d in r a p i d d u p l i c a t i on of v i d e o t a p e s u se s

a head w i t h a r a d i u s of 20 mm and a t a p e s p e e d of 2 m/s. Determine thet e n si o n t h a t must be a p p l i e d to the t a p e in o r d e r to maintain a fi l m ofair of 0.05 ^m.

ReferencesAusman, J. 8. (1959): Theory and Design of S e l f - A c t i n g . Gas-Lubricated J o u r -

n a l B e a r in g s Including Misalignment Effects, in /nêrnaMona/ ^yynposmmon Cas-LubWcoêc? Rearmys, D. D. F u l l e r (ed.). Office of Naval Research,Dept. of the Navy, Washington, pp. 161 -19 2 .Ausman, J. S. (1961): An Improved A n a l y t i c a l S o l u t io n for S e l f - A c t i n g . Gas-L u b r i c a t e d Journal B e a r i n g s of F i n i t e Length. J. BaM c En^. v o l . 83. no.2 , pp. 188-194.

Bhushan, B. (2002): 7nôa*Meôw ô JhMogy,New York, John Wiley & Sons.Blok, H., and van Rossum, J.J., (1953): "The Foil Bearing - A New Departure

in Hydrodynamic L u b r i c a t i o n , " Lu&. Em?., v . 9, pp. 316-320.Fleming, D. P., and Hamrock, B. J. (1974): Optimization of S e l f - A c t i n g Her -

ringbone Journal B e a r i n g s for aximum Stability. Proceedmys o/ ^ A e .StE^t/wêrnaMona^ Gag Rearm;? .%mpcsmm, Southampton, N. G. Coles (ed.).BHRA F l u i d E n g i ne e r in g , pp. CI-Cu.

Gross, W.A., Matsch, L.A., Castem, V., Eshel, A., Vohr, J.H., and Wildmann,M. (1980): FMa* Ft?m LubntcaMon. New York, John Wiley & Sons.



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Gunter, E. J ., Jr., Hinkle, J . G., and Fuller, D. D. (1964): The Effects ofSpeed, Load, and Film Thickness on the Performance of Gas-Lubricated.Tilting-Pad Journal Bearings. X.%.E Trans., vol. 7. no. 4, pp. 353-365.

Hamrock, B. J ., and Fleming D. P. (1971): Optimization of Self-Acting Her-ringbone Journal Bearings for aximum Radial Load Capacity. H#A 7n-%erna&!onaJ Gas .Bearmg .Symposmm,University of Southampton, paper 13.

Sternlicht, B., and Elwell, R. C. (1958): Theoretical and Experimental Analysisof Hydrodynamic Gas-Lubricated Journal Bearings. Thms. 4

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