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Research Papers: Gas Turbines: Structures and Dynamics

# A New Approach for the Stability Analysis of Rotors Supported by Gas Bearings

[+] Author and Article Information
Mohamed Amine Hassini

Mihai Arghir

e-mail: mihai.arghir@univ-poitiers.fr
Institut Pprime,
UPR CNRS 3346,
Université de Poitiers,
Chasseneuil 86962, France

The Reynolds equation is solved using the finite volumes method [13].

For clarity, the third order approximation for the cross coupled impedances is not depicted because it gives results similar to the second order.

The expressions are given only for the x direction and a similar development can be made for the ones in the y direction.

The coefficient $B2$ is not depicted since it is normalized ($B2=1)$.

For clarity, the sign $Δ$ will be dropped. The reader should keep in mind that quantities involved are perturbed values around an equilibrium position.

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received August 12, 2013; final manuscript received September 6, 2013; published online November 1, 2013. Editor: David Wisler.

J. Eng. Gas Turbines Power 136(2), 022504 (Nov 01, 2013) (11 pages) Paper No: GTP-13-1305; doi: 10.1115/1.4025483 History: Received August 12, 2013; Revised September 06, 2013

## Abstract

A simplified nonlinear transient analysis method for gas bearings was recently published by the authors (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” J. Tribol., 134(1), 011704). The method uses the fact that linearized dynamic characteristics of gas bearings, namely the impedances, can be approximated by rational transfer functions. The method gave good results if the rational transfer function approach approximated the linearized dynamic characteristics well. Indeed, each of the four complex impedances $Zαβ,α,β={x,y}$ had one or two poles depending on the order of the rational function that were used. These poles appear as supplementary eigenvalues of the extended matrix of the homogeneous system of first order differential equations describing the model of the rotor. They govern the stability of the dynamic model in the same way as the original eigenvalues do and therefore they impose non-negligible constraints on the rational function approximation of the impedances of gas bearings. The present improvement of the method overrides this problem. The basic idea is to impose the same set of poles for $Zxx$, $Zxy$, $Zyx$, and $Zyy$. By imposing this constraint, the poles are stable and the introduction of artificial instability or erratic eigenvalues is avoided. Campbell and stability diagrams naturally taking into account the variation of the dynamic coefficients with the excitation frequency can now be easily plotted. For example, the method is used for analyzing the stability of rigid and flexible rotors supported by two identical gas bearings modeled with second order rational transfer functions. The method can be applied to any bearing or seal whose impedance is approximated by rational transfer functions.

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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

Coefficients A0αβ at the equilibrium positions

Fig. 5

Coefficients A1αβ at the equilibrium positions

Fig. 6

Coefficients A2αβ at the equilibrium positions

Fig. 7

Coefficients Bk at the equilibrium positions

Fig. 8

Natural frequencies of Eq. (14) of the point mass rotor at Ω = 10 krpm

Fig. 9

Campbell diagram of the 2 dof rotor

Fig. 10

Damping ratio ν of the 2 dof rotor

Fig. 11

Rotor displacement versus time

Fig. 12

Critical mass versus rotational speeds

Fig. 13

Cosine of the angle between the fluid forces reaction and the rotor perturbation in the circumferential direction

Fig. 14

Schematic view of a rigid rotor supported by two air bearings

Fig. 15

Campbell diagram of the 4 dof rigid rotor (forward modes)

Fig. 16

Stability diagram of the 4 dof rigid rotor (forward modes)

Fig. 17

Campbell diagram of the 4 dof rigid rotor (backward modes)

Fig. 18

Stability diagram of the 4 dof rigid rotor (backward modes)

Fig. 19

The discretization used for the flexible rotor

Fig. 20

Campbell diagram of a flexible rotor (forward modes)

Fig. 21

Damping ratio (forward modes)

Fig. 22

Shape of the first three forward modes at Ω = 10 krpm

Fig. 23

Campbell diagram (backward modes)

Fig. 24

Damping ratio (backward modes)

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