Tilt stiffness and damping coefficients for finite journal bearings

Wear, 41(1977)87 0 Elsevier Sequoia

87

-102

S.A., Lausanne

- Printed

TILT STIFFNESS AND DUPING JOURNAL BEARINGS

AMALENDU

COEFFICIENTS

FOR

FINITE

MUKHERJEE

~ep~r~~e~~ of ~echunjcal 721302 (Znd~~ (Received

in the Netherlands

engineering,

Indian Ins~i~~fe of Tee~noiogy,

K~aragpur

July 1, 1975; in final form March 12, 1976)

Summary An analytical solution of Reynolds’ equation for a finite bearing with an inclined journal has been attempted. The tilt stiffness and a bearing coefficient were obtained by analysing the components of couples exerted by the fluid on the journal for small tilt and tilt rates. A transformation of the coordinate systems was adopted to separate the effects produced by tilt - change in attitude angle and change in eccentricity with length. Expressions for pressure distribution were obtained by solving Reynolds’ equation using a perturbation technique together with Fedor-type assump tions. Justification for Fedor-type assumptions is given for length-todiameter ratios greater than 0.8.

Introduction The dynamic response of elastic rotors is affected by the tilt stiffnesses and damping coefficients of the system of hydrodynamic bearings on which it is supported. Several investigators have analysed the rotor-bearing system for infinitely short bearings. A shortcoming of the narrow bearing analysis is that it predicts too high a value of load-bearing capacity. Eikuchi [l] reported the importance of tilt stiffnesses and damping coefficients in the theoretical determination of the dynamic response of an elastic system on hydrodyn~ic journal bearings. The bearing was considered to be short and thus the derivative of pressure with respect to the angular coordinate was disregarded. A similar short bearing analysis was proposed by Capeiz [ 21. Fedor [ 31 reported that the short bearing approximation, for evaluation of pressure and thus of load-bearing capacity for the journal parallel to the bearing, predicted almost twice the actual values at eccentricity ratios greater than 0.5 Fedor [4] also presented a simple procedure for finite journal bearings that provided simple expressions for the load-bearing capacity. The pressure _ function was assumed to be of the form P(@,z) = Pm(p) -xA,(z) 1

sin m/l

88

where P_(p) is the expression for the pressure of a large L/U bearing. When this form was substituted in Reynolds’ equation a set of infinite equations of three-term recurrence type was obtained. In order to reach a solution, A,(z) = -KAl(z) was taken as an initial hypothesis with K an arbitrary constant. The arbitrary constant, however, was finally evaluated using boundary conditions.

Criticism of Fedor’s solution Kuzma [5] criticised Fedor’s procedure and Donaldson [6] disagreed with Fedor’s solution when the L/D ratio was small; Fedor’s solution gives accurate results with L/D ratios of 0.8 or more. This procedure for the parallel dynamic case was used by Muster and Tolle {7] for full journal bearings.

Justification

of Fedor’s approximation

for L/D & 0.8

The accuracy of Fedor’s solution for L/D above 0.8 justifies consideration of his propo~ionality approximation. An alternative solution of Reynolds’ equation by Warner [8] was used for this. Here the procedure is by separation of variables. The pressure function was considered to be P = P&3) $-g(P)K(~) which leads to the eigenvalue d2K -----~2~~ dz2

problem

0

and d dp From the boundary conditions, an infinite set of characteristic X, is obtained and thus the pressure can be written

p=

x

aNgN(&

N=l

KN(2) (L/2)

KN

where g&3) and K,.,,(x), characteristic defined

It may be observed bolic functions.

values

I functions

corresponding

to ANaN, are

that X, is real and thus X2 > 0; hence the KN are hyper-

89

Warner noted that

f(N(z)

N= 1, 2, . . .

K:N(Ll2) formed a strongly convergent series, especially when L/D was close to or larger than 0.8. Thus a good approximation for the pressure function could be

This is the basis for the justification CS 2

of Fedor’s approximation.

Replacing

aN&@>

N=l by

2

CMsin?@

N=l

gives

This not only implies a proportionality between the fist and second coefficients of the trigonometric series but also extends it to all coefficients. Thus Fedor’s proportionality approximation is more general than Warner’s approximation, which also gives fairly accurate results at medium and large L/D ratios.

Differential

equation

Reynolds’ journal is

for a bearing with an inclined journal

equation

=$ where Ei“ = ICI + K,z

f-K’fw

as written by Kikuchi for a bearing with an inclined

- 28’) sin 0’ + 2x’ cos p’}

(1)

90

/ ,,,,

‘,,,L

I X

Fig. 1. Journal

bearing

with inclined

journal.

and fl_fl’=e’-(j

K2 =

=?z

f$sin0 -

?i _ Gc0se

(2)

J, cosf3

c + $Ysine CKl

Adopting the following transformation

of coordinates: (3)

91

eqn. (1) becomes

($+?) $i(ltK’ cc@‘)3 $ 1+-f!&](1 + K’ co@‘)35 I-

I-$ I

(1 + K’ cosp’)3 5

‘-77

= $-

{-K’(w

(1 + K’ cosp’)s >

+ ; I

1

iI

-- 28,‘) sin 0 + 2k’ cos 0’)

(4)

A point above any quantity denotes its time derivative. The boundary conditions are P(p’, L/2) = 0 and P’ @‘, - L/2) = 0 for 0 G /3’ < 277.

Reorientation of the coordinate system If the y axis is made to coincide with the line of centres on the midplane, then 0 = 0 and

Once the stiffnesses and damping coefficients are analysed in this coordinate system their evaluation in any other coordinate system becomes straightforward with the use of similarity transformations.

Analysis of stiffnesses corresponding to the 4 rotation On setting $ equal to zero (K2 = 0) and all time rates equal to zero, eqn. (4) becomes

g7]

I+; I ap, ] cosp’)35 I--n G-1 I II = dP

ap’ o+JG cosP’)3

+?i2 ~

+;

(l+ Ki cos/3’)3 2

+

(1 + Ki cosp’)s F/

(1+ Kl

(1 + Kl cosp’)3 5

I

-gK,wsinp’

+

(6)

If the 4 rotation is small, a perturbation series for P may be assumed: w

P=P,+

c

if’Pi

i= 1

Substituting this form in eqn. (6) and separating the coefficients with the same power of 5,

(7)

=

-$-

.KIwsin/3’

Equation (8) is Reynolds” equation for the parallel case written in $, z’ coordinates, and the solution for this according to Fedor [ 31 is 6PR2r(,W sin p’ (2 + Kl cm@‘) 1 _ _cosh (QdfE) C2(2 + Kf) - (1 + KI sinflT I cash (Q.L/2R) I

PO= where

g, = K,{2 + (1-+)“2}{1 When this expression is Fourier-expanded,

+ (1 -Kyly

for PO is substituted we obtain

in eqn, (9) and the right-hand

where & = J%G?I + 2x, -2ho%n) With

-3Rf

S’2+x:

C=f&

/JR&w

Bz-

C2 cosh(QL/ZR) 1

c, = zxz

2 Xn = K,(l _R;)1,2

Q

--

cm=

i-1y

Xm+1

R -Xm-1

2 a”Cm + (1 --Ky)

m = 3, 4 ..I

side

93

1 51 = (1 _ #$1/z 6

t, = 2(-l)%”

m = 2, 3 .. .

;ii, = 2(--1)“P

(1 + m(1 --Kq)--l/2)

-~;)-3/2

~(1

2(1 - KT) ho=

a

2 + Kf

m = 2, 3 . . .

Kl

= 1 + (1 - Kqy

Assuming m P = z l$(z)cos i= 0

and substituting D2B,, + $

ip

in eqn. (10) gives (D2 +2)B,

=&,sinh (1.1)

(D-n2)&+~(D2-n2+.+2)&+l+2(D2-n2--n+2)Bn__1

n = 2,3,4

KI

...

where D2 = R2a2jaz2 Assuming Bi = h& + nBi

the equation

.Bi = hBi sinh (&X/R)

for &

becomes

(12)

f_Tl L&I = {ii) with G, = Q; Tii

=

Q2 -

7'12 =

i2

K/2)(&$

+ 2)

Tii+1 = (Kl/2) (9% -ii2

+

i + 2)

and Kl

rr):,i-1 = 2

(Q,” - i2 - i + 2)

for i = 2, 3 . . .

The rest of the Tij are zero. Equation (12) can be solved to any desired accuracy. For the homogeneous part, D’hBo

+ (Kl/2) (D2 + 2)hBi = 0

94

(02 - n2) J& + (K,/2) + (K,/2)

(D2 - n2 + n + 2) hBn+ I+

(II2 - n2 -n

+ 2) hBn-l

=0

n = 2, 3 . . .

(13)

Using Fedor’s approximation f&=-l

h&I

(02 -c&

J3e = 0

gives

with

&

= C, sinh {(cr,/R)z}

&

= -C,cu, sinh {(cz2/R)z)

The boundary

conditions

&J (G/2)

= -&

(14) demand that (S/2)

and

(15) h& (&L/2) = --nBo (W2)

From this, C, and (Y~can be evaluated becomes

and the expression

1 _ cash (Qz’lR)

p = 61_1R2a (2 + Kl cos/3’)Kl sinp’ C2

(2 + Kq)(l

Z

+I7

+ Kl COS$)~

RBi COSifl' ) sinh

for the pressure

(gz’)

cosh( QL/BR)

I

t

+

i=O

+ C, (1 -cq The moment

cosp’) sinh Fz

about the X axis can be obtained L/2

M, =-R

+ . ..

ss -L/2

=;ii R

(16) as

n

Pcos(f

+ i?z’)z’dzdfi’

0

; ,,B, u2 + C=,o, -z

w

o1 (17)

The moment

about the Y axis will be

95

44, = R

7

i

-L/Z

Psin($ + tiz’).z’dz’d$

0

wK,o,

=7l

n(1 - K:)1’2

_ -i=:2,4

2 1 __2

(nBia2

+ ci+1*u3)

I

where .Bi sinhl(QL/BR)

ci = -

sinh ((r2L/2R) L3 01 = 12

+F

-

The corresponding Kx@ = M,

(g

$)tanh

stiffness components n = lICKI

(g)

are given by

KY@= MY

n = lICKI

Analysis for stiffness corresponding to the $ rotation On setting Q equal to zero and all time rates equal to zero and neglecting second and higher order terms, eqn. (4) becomes

= -E(K,

dP cospy cosp’z -

sin@ + zK2 sin/?) - 3K2

a

+ R2 a2

w’

aP

(1 + Kl co@‘) cotq.3’z I

ai3 11

where E= A solution

uuR2p c2 of the form

(19)

96

P = PO+ t: K;Pn II= 1

(20)

is assumed where POis the solution PO=

1 _ cash (QdR)

E( 2 + KI cosp)KI sir@ (2 + Kf)(l

for a parallel bearing:

I

+ K$OS@2

cash (&L/R) I

Then PI will satisfy

=-

E sin@ + 3 $

(1 + KI

i

+ R2

~~s~)~cos(~)z'

$

i

(1 + KI COS@)~cos(@z’ $

$

(21)

Again simplifying eqn. (21) and writing the Fourier component right-hand side gives (1 f KI co@) $

- 3KI sir@ $

of the

+ (1 + rc, cosp’)R2 5

,x>

Q,&

+ Dm,2

sinh

+

Dm,3z

cos

(;)I

sin m/3’

(22) where D In.1 = .E I--2& Dm.2 = -3x,

- 3h, (KIP, - %?Jl

QE R cash (Qb’2R)

and Dm,3

3Ek,, =

+ h,(K,p,v,) - (Q2/R2)l cash (QL/LR )

and E, = Zm(--1) “%“(KI(l = -m(-l)*P

77m

1 - KT)1’2] _K~j”/2 -

(i + m(

Kxfl

-K:)*“}-’

97

Assuming W P= ZZ Bi

Sini/.

i= 1

Bi = “Bi + hBi

.Bi = Eiz + J’i sinh(Qz/R) and substituting

+ HiZ cash (Qz/R)

in eqn. (22) gives

-El + K,E, = Dl,l -4Ea

- 2K, El = D2, 1

-9E,

- 2K,E, - 5&E,

(23) = Da1

.. . From the first two of these it is possible to obtain exact values of El and E2 and the remainder accurately. Similarly, from BQRH, + K,RQH,

=0

(Q2 - l)H, + 0.5K,(Q + 2)H2 = 0 (Q2 -- l)F, + 0.5Kl(Q2 + 2)F2 = Di,2 (Q2 -4)Fz

(Q2 -

4)H2

+ 0.5K,Q2F3 + 0.5Kl(Q2 - 4)F, + 2RQH2 +

KIRQHI = Di.2

+

0.5K,Q2H,

(24) +

K,RQH,

+

+ 0.5Kl(Q2 - 4)~~ = D2,3

Hi and l$ may be obtained and also HI, H2, Ha, H4, El, E2, E3, F,, F2, F3 etc. The values H,, H2, Ha, H4 etc. and Er, E2 and E3 are exact while F,, F,, F3 etc. are approximate. However, accuracy may be improved by taking more equations into consideration. Here again the homogeneous part will satisfy (D2 - l)&

+ 0.5Kl(D2 + 2) ,,l$ = 0

(D2 - l),B,

+ 0.5Kl(D2 - n2 + II + 2)hBN+1 + 0.5Kl(D2 - n2 - n + 2) X

x

h&-_l=O

n = 2, 3, . . .

98

Again adopting hBi

Fedor’s approximation

(if]

=-_nBi

(24)

and the boundary

conditions

i = 1, 2, . . .

the hBi may be fully evaluated.

The expression

for pressure becomes

X sin@

(25)

where + 4, sinh(QL/BR)

C 3,,L/2 n

+ (H,,L/2) cosh(QL/2R)

sinh ((r2L/2R)

The moment

components L/2

M,=-R

77

J-J-

-L/2

=2R

are Plz cosf3 df9 dz

0

L3 E,, z + F,,o, +

2

2n H,,o,

+

cno,

-

n2 -

n = 2.4...

1

(26)

and L/2

My = R

ss

-L/2

T!

Plz cod

de dz

0

(27)

with cJ1=

up =

0.5LR cosh(QL/2R)

= R2 sinh(QLl2R)

Q (0.5L)2

Q2

R sinh(QL/2R)

Q

- 2R

-

Q

(Jl

and (23 =

0.5RLcosh(a,L/R) a2

Corresponding K2 =-l/C

stiffnesses

-

sinh(cu,L/R) a;

again may be evaluated

by setting

99

f&l

=e

KkJI

=Mk

I

K,=-l/C

II$ =--1/c

Analysis for the damping coefficient corresponding to (b On setting $I = J/ = 0 in eqn. (4),

co@)s ~i+R’Ei(l+K~~~s~)~~j=

t](l+& 12@ =-K, c2

;i.zsinP

(28)

Let B=-

12pR2Kl

R”

C2 Assuming a solution

and substituting

it into eqn. (28) gives Kl cos~)sin~

@2+ 400

=

of the form

(2 + Kf)( 1 + Kl co@)’

Assuming m &(P, 4 = x

A&)

sin np

n=l

gives (II2 - 1) Al + 0.5 K&D2 + 2) A2 = 0

+ 0.5Kl(D2 - n2 - n + 2) + Ano Making use of Fedor’s approximation (W2PI(P) we

+

p203,

* u4

and the boundary

conditions

= 0

obtain @2 + Kl co@) sin/I ’ = (2 + Kf)(l

L sinh (cu,z/R)

+ Kl COS~~)~ ’ - ii- s~h(~2~/2~)

I

(30)

100

which leads to

BREK, M,= ?i (Z+Kf)(l-Kf) (31) REn

My=-+

(2 + K;)(l-KY)where a; = -2(1 + a,K,) 2 -aIKI

K,{2(1- K;)1'2} 'h = (1 + (1-Kf)1'2}2 E

=

WR2K,

_

c2 The damping coefficients

C,_.@=M, 1.

may be obtained

as

Cd = MY I+l,cx1

n=l/CK,

Analysis for tilt-damping corresponding to the ti rotation rate Setting $J = $I = 0 gives

Assuming p = ZPI(P) + P(P, 2) and proceeding p=-

as for tilt-damping

6/.4R2 c2

K2

2z--

KI(l+ KI cod)

corresponding

to the $J rotation

L sinh (ac,z/BR) 2

sin (a,z/2R)

rate gives

(33)

O3 =(l _ K;)1/2

M, = (34)

and

My =

12/_lR3l’i, C2

101

Fig. 2. Non-dimensional rotation stiffness us. eccentricity ratio. x corresponds to the direction perpendicular to the line of centres, I/J is the rotation about x, y corresponds to the direction of the line of centres and $ is the rotation about y.

Fig. 3. Non-dimensional

The corresponding

cxJ/

=M,

8.

rotational

damping

us. eccentricity

damping coefficients

K,=-l/C

cYJ, =My

ratio.

may be obtained as

Ii,=-l/C

Sample results are shown in Figs. 2 and 3. Acknowledgment The author would like to thank Professor J. S. Rao for advice.

102

Nomenclature radial clearance of the bearing coefficients of rotational damping eccentricity ratios of the journal (at z = 0 and at z = z, respectively) coefficients of rotational stiffness length of the bearing fluid pressure radius of the journal load on a full bearing with large L/D ratio attitude angle of the journal (at z = 0 and at z = z) coefficient of viscosity of the lubricant journal inclinations (Fig. 1) angular speed of the journal

References 1 K. Kikuchi, Analysis of unbalance vibration of rotating shaft system with many bearings and disks, Bull. Jpn. Sot. Mech. Eng., 13 (1970) 864 - 872. 2 G. Capriz and L. Galletti-Manacorda, Torque produced by misalignment in short lubricated bearings, Trans. ASME, 87D (1965) 847 - 849. 3 J. V. Fedor, Half-Sommerfeld approximation for finite journal bearings, Trans. ASME, 85D (5) (1963) 435 - 438. 4 .J. V. Fedor, A Sommerfeld solution for finite journal bearings with circumferential grooves, Trans. ASME, 82E (1960) 321 - 326. 5 D. C. Kuzma, A discussion of J. V. Fedor’s solution for finite journal bearings submitted to ASME, personal communication, 1965. 6 R. R. Donaldson, A general solution of Reynolds’ equation for a full finite bearing, J. Lubr. Technol., April (1967) 203 - 210. 7 G. C. Tolle and D. Muster, An analytical solution for whirl in a finite journal bearing with a continuous lubricating film, J. Eng. Ind., 91 (1969) 1185 - 1195. 8 P. C. Warner, Static and dynamic properties of partial journal bearings, Trans. ASME, D85(2) (1963) 247 - 257. 9 A. Mukherjee, An analytical solution of a finite bearing with an inclined journal, Wear, 29 (1) (1974) 21 - 29.