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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
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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