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J .W.Lund

Department of Machine Elements,

The Technical U nivers i ty of Denmark,

2800 Lyngby, Denmark

Review

of the

C oncept

of

Dynamic Coefficients for

Fluid Film Journal

Bearings

The developmen t of the concept of spring and dampin g coe fficients for journal

bearings is briefly reviewed. Metho ds for compu ting the coefficients are described,

and their use in rotor dynamics calculations (unbalance response, stability) is

discussed. The limitations imposed by nonlinearities on the application of the coeffi

cients is illustrated by examples.

Introduction

Th e id ea o f r ep resen t in g t h e d y n am ic r esp o n se

characteristics of a journal bearing by means of stiffness and

damping coefficients originates with Stodola

[ ]

and H umm el

[2].

Their aim was to improve the calculation of the crit ical

speed of a rotor by including the flexibility of the bearing oil

film. Concurre ntly, New kirk [3, 4] described the phenom enon

of bearing induced instability, which he called oil whip, and it

soon occurred to several investigators that the problem of

rotor stabili ty could be related to the properties of the bearing

coefficients as can be seen from numerous references in the

reference list.

In the initial stage, one of the difficulties was that the only

available solution of Reynold's equation was that of Som-

merfeld for the infinitely long bearing, sometimes modified by

a suitable end leakage factor, and tha t the only calculations of

rotor dynamics was that of the first critical speed, using the

methods of either Rayleigh or Stodola.

In the late forties, more advanced rotor dynamics calcula

tion methods were developed, and with the advent of the com

puter in the fifties, it soon became feasible at little cost to ob

tain numerical solutions of Reynold's equation and to per

form more elaborate rotor calculations.

Even so, the concept of bearing coefficients was not im

mediately accepted, probably because the load-displacement

characteristic of a journal bearing is so evidently non-linear.

Experience, however, has demonstrated the practical

usefulness of the coefficients, and modern rotor dynamics

calculations are firmly based on the concept.

Today the emphasis is on experimental measurements of the

bearing coefficients and establishing more uniform agreement

with calculated values. The need for some improvements or

refinements in the theory seem indicated, b ut the basic calcula

tion method appears sound.

The appended reference list gives some, but far from all of

Contributed by the Tribology Division of THE

AM ERICAN SOCIETY

O F

M ECHANICAL ENGINEERS and presented a t the A SM E/A SL E Joint Tr ibology

Conference, Pittsbu rgh, Pa. , Octob er 20-22 , 1986. M anuscript received by the

Tribology Division M arch 14, 1986. Paper N o. 86-Trib-48.

BEAR ING

CENTER

STAT IC EQU IL IBR IUM

.LOCUS CURVE

• x

F i g . 1 C o o r d in a t e s y s t e m

the papers that mark the development of the concept of bear

ing coefficients. L it t le attem pt has been made of a historical

review, and there is no systematic scheme behind the selection

of the papers. The cited works, however, are sufficiently

representative to allow compiling a com prehensive l ist by com

plementing the present reference list by those of the papers.

Analys is

In an x-j '-coordinate system with origin in the bearing center

and the x-axis in the static load direction (see Fig. 1), the reac

tion forces from the lubricant film are given by:

/• LII |> e

J

LI I

Jfl

cos(T]

\RdQdz

sinflj

(1)

where /? is the pressure in the fi lm, R is the journal radius, L is

the axial length, z is the axial coordinate, and 9 is the cir-

Journal of Tribology

JANUARY 1987, Vol. 10 9/3 7

Copyright © 1987 by ASME

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cumferential angular coordinate, measured from the negative

x-axis. The film extends from 0, to d

2

where both angles may

be functions of z.

The film pressure derives from Reynold's equation:

dp

h* dp

Rd d Vl2 / i Rdd J dz V 12^7 dz

+

U

) -

ViQ-

dh dh

~W

2)

where / is t ime, n is the lubricant viscosity, Q is the jo urna l

angular speed of rotation, and h is the film thickness:

h = C + xcosB + ysind

3)

C is the radial clearan ce, and x and y are the coordinates of the

journal center.

It appears that the reaction forces are functions of x and y,

and of the instantaneous journal center velocities, x and y

( dot indicates t ime derivative). Hence, for small amplitude

mot ions,

Ax

and

Ay ,

measured from the static equilibrium

position (x

0

, y

0

), a first orde r Taylor series expa nsion yields:

+ K

xx

Ax+K

xy

Ay + B

xx

Ax+B

xy

Ay

F

y

= F

yo

+ K

yx

Ax+K

yy

Ay + B

yx

Ax+B

yy

Ay

4)

where the coefficients are the partial derivatives evaluated at

the equilibrium position:

K

r]

EL)

dy Jo

B

r

\ dy Jo

5)

and analogously for the remaining coefficients.

Because (x

0

, y

0

) is the equilibrium position, then F ̂ = 0

while F ̂ equals the static load, W.

In the literature the coefficients are often computed directly

by numerical differentiation. While this may be adequate for

practical purposes, the inherent numerical inaccuracy of the

method can be eliminated by employing a perturbation solu

tion. Thus, equation (3) may be written as:

where:

h = h

0

+ Ah

h

0

= C+x

0

cos8+y

0

sinf?

Ah = A xcos0 + A ysinfl

dh

~dT

= A icosS + AysinO

(6)

(7)

(8)

(9)

The perturbation in film thickness gives rise to a similar per

turbation in the fi lm pressure:

P =Po +

A

P

(10)

where:

A/7

=

p

x

'Ax+p

y

-Ay+p

x

-Ax+p;'Ay (11)

By substituting equation (6) to (11) into Re ynold's equatio n,

equation (2), and by retaining first order terms only, five

equations are obtained:

R{ p

0

) = V2Q

dh

0

dd

••

1

/2O (-;c

o

sin0+.)>

o

cos0)

R{ p

x

h

0

dd

, K dp

0

12ji Rdd

d / cos0\

Rdd \ h

n

J

VittsmO

(12)

(13)

R{ p

y

} = -1-

sin0

^7

<Ati

/ i

3

0

dh

0

dd

dPo

12/t Rdd Rdd \ h

f s i n 0 \

\~hTJ

+ ViUcosd (14)

R [p

x

} =

cosO

R {p'

y

} = sin0

where the left-hand side operator is:

d ( h\ d \ d

R[

/ dz V 12u dzJ

(15)

(16)

(17)

Rd6\l2n Rdd/ dz V 12/i d z /

The boundary conditions are that the pressure is zero at the

edges

{p

0

=

Ap

= Q a t

z

= ±

ViL)

and equals the feeding

pressure at any supply grooves.

When film rupture occurs in the divergent fi lm region, the

location of the trail ing end bounda ry is governed by the condi

tion of zero pressure gradient, requiring some iterative pro

cedure to determine the actual boundary curve. If (0

O

, z

0

) is a

point on the curve, the pressure in a neighboring point is, to

the first order:

p(6

0

+ A6,z

0

+ Az) =p(6

0

,z

0

) + ( - ^ - )

o

A8

( -^ - )

Q

A Z

=Po(.e

Q

,z

0

) + Ap(d

0

,z

0

)+ (-jf)

Q

Ad+ (-j£-)

0

A z (18)

With p

0

(6

0

, z

0

), (dp

0

/dd)

0

, (dp

Q

/dz)

0

being zero, and by

assu m in g th a t t h e p ressu re o n t h e b o u n d ary , p(8

0

+

A0,Zo + A z), is zero also und er dynamic conditions then the

boun dary con dition for the perturbed pressu re is that A p is

zero on the statically determined c urve. In case of film ruptu re

(film contraction) ahead of the leading edge of the film, the

location of the boundary curve is governed by flow

equilibrium such that the pressure gradient is not zero. Even

so ,

the condition of Ap being zero on the boundary curve is

sufficiently accurate for most purposes.

By combining equations (1), (4), (10), and (11) it is found

that:

K„

K

v

,

f

L /2 j

e

2

J -111 J 8,

X

cosf?

sing

VRdddz

(19)

and analogously for the remaining coefficients.

Because they derive from self-adjoint operators, the cross-

coupling damping coefficients are equal:

B

yx=

B

xy (20 )

The damping matrix, therefore, is symmetric and possesses

principal directions. This is not true for the stiffness matrix w h

ere the cross-coupling stiffness coefficients are unequal:

K

yx*K

xy

(21)

Thus, the stiffness matrix is nonconservative as is readily

demonstrated by considering a closed whirl orbit with angular

whirl frequency co:

Ax= \Ax

I cos pit + 4>

x

)

(22)

Ay= \Ay\sin(wt

+ 4>

y

)

where <j>

x

and <j>

y

are phase angles. The energy dissipated over

one cycle is:

Ediss = ™(Bxx\Ax\

2

+B yy\Ay\

2

- {B

xy

+ B

yx

) I Ax I • IA^ I sin(<k - 4>

y

)

- ir K

xy

-K

yx

) \Ax I . I Ay

I

cosfo , - </>,)) (23)

With K

xy

po sitive, the cross-c oupling stiffness coefficients

may provide negative damping or, in other words, they can

supply energy to the motion. The last term may also be written

as:

ir K

xy

- K

yx

){ - I Ax j I

2

+ I Ax

b

\

2

) (24)

where the ell iptical whirl orbit has been decomposed into two

circular orbits, one with forward whirl direction and

amplitude \Ax

f

\, and one with backw ard whirl direction and

3 8 / Vo l . 1 0 9 , JA NUA RY 1 9 8 7

T r a n s a c t i o n s o f t h e A S M E




amplitude \Ax

b

I. Thus i t is the forward whirl com pone nt

which is responsible for the negative damping.

It is customary to analyze the stability of a bearing by

assigning a mass M to the journal such that the equations of

motion become:

MAx

+

B

xx

Ax

+

K

xx

Ax

+

B

xy

Ay

+

K

xy

Ay

=

0

MAy + B

yy

Ay + K

yy

Ay + B

yx

Ax + K

yx

Ax = 0

These equations admit of solutions for Ax and Ay of the form

e

5

' where:

s = \ + iu (i = ̂ l ) (26)

For a nontrivial solution, the eigenvalues s are the roots of

the determina nt of the coefficient matrix. A t the threshold of

instability, X is zero whereby equation (25) takes on the form:

of frequency. For a plain journal bearing B

mm

is always

positive, while B

min

is negative for frequencies smaller than co

0

(for o>

= oi

0

,

B

min

equals zero and

K

min

equals

K

0

).

Hence, if

the lowest natural frequency of the system, co

nat

, is less than

co

0

, self-excited whirl can be sustained. The speed, fi

0

, at which

instability sets in, may then be expressed as:

Q

n

35)

( o )

0

/ Q )

0

A typical value for the whirl frequency ratio , (co

0

/Sl)

0

, is

around 1/2 from which stems the rule-of-thumb that a rotor

becomes unstable when the speed reaches twice the first

critical speed (~co

na t

) .

(Z

xx

-u

2

M)

where:

(Z

yy

-u

2

M)

--K

rr

+ iuB

x

Ax

Ay

= 0

(27)

(28)

and similarly for the other impedances.

For a chosen static equilibrium position (Sommerfeld

number), the values of the coefficients are given, and setting

the determina nt equ al to zero determines that value of M , call

ed M

crit

> which makes the specified eq uilibrium position

unstable. The solutions for M

crk

and the associated whirl fre

quency,

O

= CO

0)

are:

6

M

c r i t =

*^xx yy **^yy**xx *^xy yx ^yx&x.

^ xx • **yy

i

K

xx — « o ) ( ^ v v

_

*o ) - K

xy

K

K

0

0 =

R R

^xx^yy

' ^xy^yx

(29)

(30)

In order to establish on which side of the threshold the be ar

ing becomes unstable , the determ inant, A , may be expande d

around the threshold value as:

A s A o + f — ) >ds+(-4^r) -dM (31)

0

Vidco/o \ dM Jo

With A and A

0

equal to zero, the equation may be solved:

(-

dM

( -

dX

dM

dw

~ d ~

/ o

=

\ dw )ol\ dM / o

from which

dX

ViufcBxx + B

yy

){B

xx

B

yy

-B

xy

B

yx

)

<ft\ Viu%(B

xx

+

B

yy

\B

xx

B

yy

-

\dM Jo ufcB

xx

+ B

yy

)

2

K

2

0

+ {K^K^-

K

xy

K

yx

-K\f

(33)

J

yyf

,v

0 ' \**xx -yy **xy

x

*yx

This value is positive which means that an increase, dM , in

journal m ass above the crit ical value results in a positive value

of X or, in other wor ds, causes the bearing to become unstab le.

In equation (27) the term - w

2

M represents the rotor im

pedance at the journal . It may be thought of as being opposed

by some equivalent bearing impedance Z which, then, is a

solution to the determinantal equation:

Z= a ( Z „ + Z

y

,)±J—(Z„ - Z

yy

)

2

+

Z

xy

Z

yx

•^max ~~ ^m ax •

(

™ m a i

= K

mm

+ iwB„

(34)

The effective stiffnesses, K

mm

and -K

min

, and the effective

damping coefficients, B

max

and B

min

, are of course functions

Dis c us s ion

The preceding stability analysis applies only to a rigid rotor

which is symmetric and supported in two identical bearings.

Hence, the results are of limited usefulness, and in general it

can be quite misleading to rank various bearing types for

stability on the basis of the critical mass.

A s seen from equation (29), the crit ical mass value is a

measure of the stiffness of the bearing, but an increase in bear

ing stiffness does not ensure that the lowest natural frequency

is raised to improve the instabili ty threshold speed.

Similarly, equation (30) suggests that bearings with a low

whirl frequency ra tio may be more stable. For a flexible roto r,

however, it is found that it is the whirl frequency itself, equal

to the lowest natural frequency, which stays unchanged as the

rotor becomes unstable, not the whirl frequency ratio .

Furthermore it should be noted that associated with the

calculated instabili ty threshold is a characteristic whirl orbit ,

obtained as the solution for

Ay/Ax

from equation (27) with

the determinant being zero. For a flexible, non-symmetric

rotor, the modeshape will not conform to the characteristic or

bit, and thereby the effective stiffness and damping stability

properties of the bearing change.

For a practical rotor there is no real substitute for a stabili ty

calculation in which the roto r, with i ts bearings represented by

their coefficients, is modeled as close to the actual design as

feasible. Such calculations are today standard procedure when

designing high speed machin ery. The results are in sa tisfactory

agreement with experience, considering the uncertainties and

often poorly established tolerances on bearing geometry,

clearance, and pedestal st iffness.

A different type of calculation, bu t with the same rotor-

bearing-system model, is the one for the response to mass un

balance. It is used to check the unbalance sensitivity of the

rotor and, also, to determine the damped crit ical speeds.

It is inherent in the definition of the bearing stiffness and

damping coefficients t hat , mathe matically, they are only valid

for infinitesimal amplitudes. Comparisons with exact solu

tions, however, prove that for practical purposes the coeffi

cients may be employed for amplitudes as large as, say, forty

percent of the clearance.

A n example is shown in Fig . 2 which is for a two-axial

groove journal bearing with a length-to-diameter ratio of

X

A.

The journal is forced to whirl synchronously by a rotating un

balance force, equal to half the static load, while the inertia of

the journal is ignored.

The elliptical whirl orbit, calculated on the basis of the eight

coefficients belonging to the indicated static equilibrium posi

tion, is compared to the whirl orbit obtained from a true

nonlinear solution. It is found that the amplitudes, whith a

maximum value of 38 percent of the clearance, are in close

agreement, but, as expected, there are also deviations.

First , the center of the orbit moves closer to the bearing

center to satisfy overall dynamic equilibrium:

Journal of Tribology

J AN U AR Y 198 7, Vo l . 1 0 9 / 3 9




-0.2 -0.1 R 0.1 0.2 0.3 O.t 0.5 0.6 0.7 OB 0.9 10

STATIC LOAD DIRECTION

F i g .

2 U n b a l a n c e w h i r l o r b i ts . C o m p a r i s o n b e t w e e n l i n e a r a n d

n o n l in ear an alysis .

F

r

ds=W

F

y

ds = 0

(36)

where s is the arc length along the path of the orbit . Second,

the shift in orbit center is accompanied by a change in orbit

orientation which affects the agreement in phase angle. Fin-

nally, it is evident that the non-linear orbit contains higher

order harmonics which of course are absent in the l inear

solution.

It should be emphasized, however, that in the matter of

greatest importance, namely the size of the amplitude, the

agreement is close. This holds true in the range of interest in

practice where amplitudes larger than, say, 40 percent of the

clearance would rarely be tolerated.

For even larger amplitudes, the various phenomena

associated with non-linear vibrations become evident. A s an

example, Fig. 3 i l lustrates the existence of multiple so lutions in

certain operating regions. The diagram is computed for the

idealized case of an unloaded, infinitely short journal bearing

where the jour nal , with mass M , whirls synchronously in a cir

cular orbit whose radius, normalized with respect to the radial

clearance C, is given by the ordinate.

The curves in the diagram are labeled by the dimensionless

unbalance mass eccentricity

p,

normalized by

C.

The abscissa

gives the dimensionless journal mass m = CMQ,

2

/oW where <r

is the short bearing Sommerfeld number:

f iURL / R

W

( 4 - ) ( 4 - )

37)

Hence, for a particular rotor and bearing, the abscissa is

simply proportional to the rotational speed.

It is seen that, for some given mass un balan ce, there will be

a speed beyond which more than one solution is possible and

where jump phenomena may occur . The l inear ized so lu t ion

with stiffness and damping coefficients agrees well with the

small amplitude orbit , but i t is of course unable to predict the

concurrent orbits with larger amplitudes.

Similarly the previously discussed stability analysis, based

on the liniearized coefficients, is concerned only with the

stabili ty of the static equilibrium and the inception of whirl ,

and it can not be used to study whether that whirl orbit may

actually be contained and stable (l imit cycle). Furthermore,

while the l inear analysis predicts that an unloaded journal

bearing is inherently unstable, the stabili ty curve in Fig. 3

demonstrates that stable synchronous whirl orbits may exist

10 10* 10'

t

DIMENSIONLESS JOURNAL MA SS : i^ -^ T?

F i g .

3 C i rcu lar u n b alan ce wh i r l o rb i ts fo r u n lo ad e d , sh o r t jo u rn a l b ear

ing with r ig id rotor

for p greater than 0.13 or, in other words, the bearing can be

stab i l ized by adding unbalance.

The nonlinear behaviour, however, is rarely of much con

cern in practice, and for practical purposes the instabili ty

threshold c ompute d on the basis of the l inearized coefficients

should be adopted as the design criterion. Past the threshold

speed, self-excited whirl takes place in one form or another,

and although numerous cases are on record of safe operation

in this region, at least as many cases of uncontrollable whirl

can be cited.

The threshold based on the coefficients proves in general,

therefore, to be a conservative estimate. There remain,

however, some basic questions about applying Reynold's

equation. The major problem is the physics of fi lm rupture

which has yet to be clarified, and which makes uncertain what

the proper boundary condi t ions are under dynamic load . A s a

curiosity, another example is the question of the thermal con

ditions prevailing under synchronous whirl when the orbit en

circles the bearing center.

Finally, the four stiffness coefficients can be derived direct

ly from the static equilibrium locus curve. It is unfortunate,

therefore, that experimental measurements of the curve all too

frequently deviate appreciably from the theoretical curve. The

measurements are admittedly not easy to make, and the ex

perimental tolerances are difficult to ascertain, but enough

results have been published to indicate that the discrepancies

are real .

Under those conditions, experimental measurements of the

coefficients (typically by applying a vibratory force and

measure the resulting response) can not be expected to agree

well with the theoretical values. Th e experiments are no longer

a test of the validity of the dynamic analysis; rather, they

become indirect evidence of discrepancies in the more fun

damental solution of Reynold's equation for static conditions.

Conc lus ion

The concept of stiffness and damping coefficients for jour

nal bearings has proven very fruitful, and modern rotor

dynamics calculations for unbalance response, damped

natural frequencies, and stabili ty are based on this concept.

The theoretical limitation of small amplitudes is of little im

portance in practice, but the analysis sti l l needs imp rovem ents,

for example in refinements in the model for fi lm rupture.

In recent years, extensive experimental programs have been

carried out to measure the bearing coefficients, not just in the

laboratory, but also for large industrial bearings. While many

results show good agreement with the theoretical values, cases

are sti l l reported of discrepancies that should be resolved.

4 0 / V o l .

109 , JAN U AR Y 1987

T r a n s a c t i o n s o f t h e A S M E




The question of the influence from therma l and elastic

deformations needs further consideration as do also the prac

tical problems

of

man ufacturing and operating tolerances

on

bearing geometry, clearance, and lubricant viscosity.

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Some Problems of Rotor Dynamics,

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the

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Y., A

Theory

of Oil

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J ou r na l o f T r i b o l o g y

J AN UA RY 198 7 , V o l . 1 0 9 / 4 1

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