APPENDIX 10

Fluid Circumferential Average Velocity Ratioas a Journal Eccentricity Function Based

on Lubrication Theory

A journal of radius R rotates at a constant rotational speed O within a cylindricalannulus of bearing clearance. The radial clearance, c, is small in comparison to R. Theannular space between the journal and bearing is filled with an incompressible fluid withuniform dynamic viscosity . The axial flow is assumed to be negligible. The external verticalforce with magnitude P is applied to the journal causing its center to displace from thebearing center. It is assumed that cavitation in the bearing does not occur. The journal isorbiting with angular speed ! and amplitude A around its displaced equilibrium position e.

The fluid film thickness h varies with instantaneous angular position around the annularspace. For cR , the relation is as follows (Figure A10.1):

h ,t c e cos A cos !t A10:1

Under the assumption of laminar viscous steady-state flow, with no pressure variationacross the thickness of the film, the only possible flow patterns that satisfy the requirementsof fluid dynamics in a uniform channel of thickness, h, are combinations of two basic

Figure A10.1 Fluid film thickness variations when the center of the rotating journal Os is displaced and the rotor

is orbiting with the angular velocity O.

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patterns shown in Figure A10.2. The first pattern is linear due to journal rotation. Thesecond pattern has a parabolic shape, and is associated with the journal radial displacementfrom the concentric position. The following relationship determines the fluid velocity u, thefluid volume flow rate Q, and gradient of the pressure p:

u R!y=h C y=h y2=h2 A10:2Q R!h=2 Ch=6 A10:3

@p

@ R @2u

@y2 2C=h2 A10:4

The parameter, C, represents the flow velocity, and it has to be determined from the flowcontinuity requirements:

@Q=@ 0 A10:5

and pressure periodicity:

p p 2 A10:6

The condition (A10.5) yields the following differential equation:

@Q

@ 6

3RO C @h@ h @C

@

0 A10:7

which has the solution

3R! C h C1 const: A10:8

The constant of integration, C1, has to be calculated from the condition (A10.6).Eq. (A10.4) provides:

dp

d 2R C1

h3 3R!

h2

and further

p 2 p 0 2R C1Z 20

d

h3 3R!Z 20

d

h2

0 A10:9

Figure A10.2 Components of the laminar flow in the journal/bearing clearance with uniform pressure across the

clearance (L length of bearing).

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Taking Eq. (A10.1) into consideration and performing integration provides the constantC1, which introduced to Eq. (A10.8) provides, in turn, the constant C:

C 3R! 2 1 "2

2 "2

1

1 " cos 1

6 sinRA 1 " cos O

1 "22 "2

cos !t 1 " cos

6" cos!t

2 "2 2" #

"! cos !t 32 "2 cos!t

A10:10

where " e=c is the eccentricity ratio.The pressure distribution is obtained by integrating the pressure gradient Eq. (A10.4).

The resultant fluid force F acting on the journal due to these pressures has the followinghorizontal, Fh, and vertical, Fv, components:

Fh 12R3A

c3 1 "2 3=2 2 "2 1 "2 O 2 "2 ! sin!t A10:11

Fv 12R3A"1

c3 1 "2 3=2 2 "2 1 "2 O 2 "2 ! 3"2 "2O

2 "2 !

cos!t

The total work performed on the journal by these forces during one orbiting cycle, i.e., theenergy per cycle DE transmitted to the orbiting is

DE Z 2=O0

Fhdh Z 2=O0

Fvd A10:12

From the projections of displacements on the horizontal and verticalaxes are:

dh A! sin!t dt, d A! cos!t dt A10:13

Taking equations (A10.11) and (A10.13) into account, Eq. (A10.12) becomes

DE 122R3A2

c2 1 "2 3=2 2 "2 "4 2"2 4

2 "2 O 4 "2

!

For the rotational speed

O54 "2 2 "2 "4 2"2 4 ! A10:14

DE50 and the fluid film forces are stabilizing. For

O44 "2 2 "2 "4 2"2 4 ! A10:15

FLUID CIRCUMFERENTIAL AVERAGE VELOCITY RATIO 1029

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Figure A10.1,

the fluid film forces are destabilizing the journal. The neutral stability occurs when

O 4 "2

2 "2

"4 2"2 4 ! A10:16

which yields the fluid circumferential average velocity ratio as a function of eccentricity:

l !O "

4 2"2 44 "2 2 "2

3 1 "2 29 1 "2 2

A10:17

This relationship is presented in Figure A10.3. As can be seen, the fluid circumferentialaverage velocity ratio is a decreasing function of the eccentricity ratio.

Figure A10.3 Fluid circumferential average velocity ratio as a function of the journal eccentricity ratio.

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Table of ContentsAPPENDIX 10: Fluid Circumferential Average Velocity Ratio as a Journal Eccentricity Function Based on Lubrication TheoryAPPENDIX 1: Introduction to Complex NumbersAPPENDIX 2: Routh-Hurwitz Stability CriterionAPPENDIX 3: Rotor Lateral Motion Forced SolutionsAPPENDIX 4: Relations Between Bearing Dynamic Coefficients In Two Fixed FramesAPPENDIX 5: Gyroscopic Rotor Responses to Synchronous and Nonsynchronous Forward and Backward PerturbationAPPENDIX 6: Basic Trigonometric RelationshipsAPPENDIX 7: Couette FlowAPPENDIX 8: Matrix Calculation ReviewAPPENDIX 9: Numerical Data for Rotor Lateral/Torsional Free VibrationsGlossary