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Research Papers: Gas Turbines: Structures and Dynamics

# A Simplified and Consistent Nonlinear Transient Analysis Method for Gas Bearing: Extension to Flexible Rotors

[+] Author and Article Information
Mohamed Amine Hassini

EDF R&D,
1 avenue du Général de Gaulle,
Clamart 92141, France
e-mail: mohamed-amine.hassini@edf.fr

Mihai Arghir

Pprime Institute,
UPR CNRS 3346,
Université de Poitiers,
11 Bd Pierre et Marie Curie,
Futuroscope Chasseneuil,
Poitiers 86962, France
e-mail: mihai.arghir@univ-poitiers.fr

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received November 18, 2014; final manuscript received January 21, 2015; published online February 25, 2015. Editor: David Wisler.

J. Eng. Gas Turbines Power 137(9), 092502 (Feb 25, 2015) (10 pages) Paper No: GTP-14-1629; doi: 10.1115/1.4029709 History: Received November 18, 2014

## Abstract

A simplified, new method for evaluating the nonlinear fluid forces in air bearings was recently proposed (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” ASME J. Tribol., 134(1), p. 011704). The method is based on approximating the frequency dependent linearized dynamic coefficients at several eccentricities, by second-order rational functions. A set of ordinary differential equations is then obtained using the inverse of Laplace transform linking the fluid forces components to the rotor displacements. Coupling these equations with the equations of motion of the rotor leads to a system of ordinary differential equations where displacements and velocities of the rotor and the fluid forces come as unknowns. The numerical results stemming from the proposed approach showed good agreement with the results obtained by solving the full nonlinear transient Reynolds equation coupled to the equation of motion of a point mass rotor. However, the method (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” ASME J. Tribol., 134(1), p. 011704) requires a special treatment to ensure continuity of the values of the fluid forces and their first derivatives. More recently, the same authors (Hassini, M. A., and Arghir, M., 2013, “A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings,” ASME Paper No. GT2013-94802) showed the benefits of imposing the same set of stable poles to the rational functions approximating the impedances. These constrains simplified the expressions of the fluid forces and avoided the introduction of false poles. The method in (Hassini, M. A., and Arghir, M., 2013, “A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings,” ASME Paper No. GT2013-94802) was applied in the frame of the small perturbation analysis for calculating Campbell and stability diagrams. This approach also enhances the consistency of the fluid forces approximated with the same set of poles because they become naturally continuous over the whole bearing clearance while their increments were not. The present paper shows how easily the new formulation may be applied to compute the nonlinear response of systems with multiple degrees of freedom such as a flexible rotor supported by two air bearings.

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## References

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## Figures

Fig. 1

Schematic view of a journal air bearing

Fig. 2

Stiffness coefficients and their approximation with rational functions (centered rotor)

Fig. 3

Damping coefficients and their approximation using rational functions (centered rotor)

Fig. 4

Large displacement (the rotor is in equilibrium position at (x0, y0))

Fig. 5

Static forces in the cylindrical frame

Fig. 6

Coefficient A0αβ in the cylindrical frame

Fig. 7

Coefficient A1αβ in the cylindrical frame

Fig. 8

Coefficient A2αβ in the cylindrical frame

Fig. 9

Coefficients B1 and B2

Fig. 10

Orbit of the rotor

Fig. 11

Fluid forces versus time

Fig. 12

FFT of the displacement in the X direction

Fig. 13

FFT of the displacement in the Y direction

Fig. 20

Poincaré diagram in the XZ plane

Fig. 19

FFT of displacement in the X direction Ω (=10 krpm)

Fig. 18

Orbit of the rotor at bearing position and at shaft center (Ω=10 krpm)

Fig. 17

FFT of displacements in the X direction (Ω=20 krpm)

Fig. 16

FFT of the displacements in the X direction Ω (=15 krpm)

Fig. 15

Orbit of the rotor Ω (=15 krpm)

Fig. 14

The discretization used for the flexible rotor

Fig. 21

Poincaré diagram in the YZ plane

## Errata

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