Automatic Balancing of a Rigid Rotor with Misaligned Shaft

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231-236

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October 2006

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© 2006 Trans Tech Publications Ltd. All Rights Reserved

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[7] Genta G. Delprete C. and Busa E., Some considerations on the basic assumptions in rotordynamics, Journal of Sound and Vibration 227(3) (1999) 611-645. APPENDIX The explicit forms for the kinetic energy T , potential V and Rayleigh's dissapation function F which are derived in Section 2 are given here T = 1 2Ix � ˙φ cos θ cos ψ + ˙θ sin ψ�2 + 1 2Iy �− ˙φ cos θ sin ψ + ˙θ cos ψ�2 + 1 2Iz � ˙φ sin θ + ˙ψ�2 +1 2M � ˙X2 + ˙Y 2 + ˙Z2� 1 2 2n+2X i=1 mi� h ˙X + εi � − sin θ ˙θ cos (ψ + αi) − cos θ sin (ψ + αi) � ˙ψ + ˙αi�� + zi cos θ ˙θi2 + h ˙Y + εi� cos θ ˙θ sin φ cos (ψ + αi) + sin θ cos φ ˙φ cos (ψ + αi) − sin θ sin φ sin (ψ + αi) � ˙ψ + ˙αi� − sin φ ˙φ sin (ψ + αi) + cos φ cos (ψ + αi) � ˙ψ + ˙αi�� + zi �sin θ ˙θ sin φ − cos θ cos φ ˙φ � i2 + h ˙Z + εi� − cos θ ˙θ cos φ cos (ψ + αi) + sin θ sin φ ˙φ cos (ψ + αi) + sin θ cos φ sin (ψ + αi) � ˙ψ + ˙αi� + cos φ ˙φ sin (ψ + αi) + sin φ cos (ψ + αi) � ˙ψ + ˙αi�� + zi �− sin θ ˙θ cos φ − cos θ sin φ ˙φ � i2� , (A-1) V = 1 2 2X j=1 n kxj �X + sin θ ˜lj�2 + kyj �Y − cos θ sin φ ˜lj�2 + kzj �Z − (1 − cos θ cos φ) ˜lj�2 o , (A-2) F = 1 2 2X j=1 n cxj � ˙X + cos θ ˙θ ˜lj�2 + cyj � ˙Y + �sin θ ˙θ sin φ − cos θ cos φ ˙φ � ˜lj�2 czj � ˙Z − �sin θ ˙θ cos φ + cos θ sin φ ˙φ � ˜lj�2 o + 1 2 2n+2X i=3 cb ˙αi. (A-3)

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