APPENDIX 4

Relations Between Bearing Dynamic CoefficientsIn Two Fixed Frames

The relationship between bearing dynamic coefficients in two different fixed frames canbe given in the following.

Components of bearing force F, involving stiffness and damping terms in XY referenceframe, as shown in Figure A4.1, can be expressed as

FxFy

Kxx Kxy

Kyx Kyy

xy

Dxx Dxy

Dyx Dyy

_xx_yy

A4:1

The same bearing forces can also be expressed in XrYr reference frame as

FxrFyr

K11 K12

K21 K22

xryr

D11 D12

D21 D22

_xxr_yyr

A4:2

Note that

FxrFyr

T Fx

Fy

A4:3

where

T cos sin sin cos

A4:4

Figure A4.1 Bearing fluid force components in two reference frames.

999

2005 by Taylor & Francis Group, LLC

It can be seen that T 1 T T. The displacements in the two different frames can be

xy

T 1 xr

yr

A4:5

Differentiating Eq. (A4.5) with respect to time results in

_xx_yy

T 1 _xxr

_yyr

ddt

T 1 xryr

A4:6

Since is independent of the time, the above second term vanishes (it would be non-zero should reference frame XrYr rotate relative to XY reference frame). Comparison ofEqs. (A4.1) and (A4.2) after substitution of Eqs. (A4.3) to (A4.6) into Eq. (A4.1) results in

K11 Kxx cos2 Kyx Kxy

sin cos Kyy sin2 K12 Kxy cos2 Kxx Kyy

sin cos Kyx sin2

K21 Kyx cos2 Kxx Kyy

sin cos Kxy sin2 K22 Kyy cos2 Kyx Kxy

sin cos Kxx sin2

A4:7

and

D11 Dxx cos2 Dyx Dxy

sin cos Dyy sin2 D12 Dxy cos2 Dxx Dyy

sin cos Dyx sin2

D21 Dyx cos2 Dxx Dyy

sin cos Dxy sin2 D22 Dyy cos2 Dyx Dxy

sin cos Dxx sin2

A4:8

It can be seen that, in general, elements of stiffness and damping matrices vary withaxis orientation. There exists one exception: when KxxKyyK1, Kxy KyxK2, andDxxDyyD1, DxyDyxD2. In this case

K11 K12

K21 K22

" #

Kxx Kxy

Kyx Kyy

" #

K1 K2

K2 K1

" #

D11 D12

D21 D22

" #

Dxx Dxy

Dyx Dyy

" #

D1 D2

D2 D1

" # A4:9

Using Eqs (A4.1), (A4.7) and (A4.8), it can then easily be proved that

lO Kxy KyxDxx Dyy

K12 K21D11 D22 A4:10

where l is fluid circumferential average velocity ratio and O is rotational speed. Thusthe parameter lO, the angular speed at which the fluid force rotates, is invariant to therotation of the stiffness and damping matrices, i.e. it is invariant to the assigned coordinatesystem.

1000 ROTORDYNAMICS

2005 by Taylor & Francis Group, LLC

related by (see Appendix 8):

Table of ContentsAPPENDIX 4: Relations Between Bearing Dynamic Coefficients In Two Fixed FramesAPPENDIX 1: Introduction to Complex NumbersAPPENDIX 2: Routh-Hurwitz Stability CriterionAPPENDIX 3: Rotor Lateral Motion Forced SolutionsAPPENDIX 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 VibrationsAPPENDIX 10: Fluid Circumferential Average Velocity Ratio as a Journal Eccentricity Function Based on Lubrication TheoryGlossary