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Hydrodynamic Lubrication

# A Simplified Nonlinear Transient Analysis Method for Gas Bearings

[+] Author and Article Information
M. Amine Hassini1

Institut Pprime, UPR3346 CNRS, Université de Poitiers, Francemohamed.amine.hassini@univ-poitiers.fr

Mihai Arghir

Institut Pprime, UPR3346 CNRS, Université de Poitiers, Francemihai.arghir@univ-poitiers.fr

Rational functions of second order proved a posteriori to be a good compromise between accuracy of fit and computational effort.

$bkαβ$ and $akαβ$ are, respectively, the residues and the poles of the approximated impedance $Zαβ$ (see Eq. 2).

The inverse Laplace Transform of a constant $c$ is: $Λ-1(c)=cδ(t)$ where $δ(t)$ is Dirac function.

A function $f(t)$ is causal if $f(t)=0$ for $t<0$.

The development is limited to the X direction, the same reasoning being used in the Y direction for the components $Δfyx$ and $Δfyy$ and their respective derivatives.

Since $Dxx$ and $Dxy$ are polynomials in C with real coefficients, their zeros comes in conjugate pairs. Hence, if the imaginary part of one of the poles is complex (with non-null imaginary part) then the impedances shares automatically two poles.

1

Corresponding author.

J. Tribol 134(1), 011704 (Feb 24, 2012) (12 pages) doi:10.1115/1.4005772 History: Received July 09, 2011; Revised December 15, 2011; Published February 21, 2012; Online February 24, 2012

## Abstract

The traditional small perturbation method is successfully used for linear dynamic analysis of gas bearings but excludes any nonlinear study. Investigating large displacements requires the evaluation of the nonlinear aerodynamic forces in the thin film. To avoid solving the unsteady compressible thin film fluid equations, we propose a method based on the use of frequency dependent dynamic coefficients and on the rational function approximation of the resulting impedances. Calculating impedances for several eccentricities enables mapping the full dynamic behavior of the bearing. A set of ordinary differential equations is then developed by using the inverse of Laplace transform. The equations of motion of the rotor are subsequently solved numerically with local linearization at each time step. The numerical results obtained by using impedances are in good agreement with the reaction forces obtained by solving the full nonlinear transient Reynolds equation.

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## Figures

Figure 1

Dynamic coefficients of the 1D parallel slider

Figure 2

Direct and cross coupling stiffness coefficients versus excitation frequency (L/D = 1, C/R = 1.1·10−3 , Ω = 25 krpm)

Figure 3

Direct and cross coupling damping coefficients versus excitation frequency (L/D = 1, C/R = 1.1·10−3 ,Ω = 25 krpm)

Figure 4

Stiffness coefficients versus excitation frequency (centered rotor) (L/D = 1, C/R = 1.1·10−3 , Ω = 25 krpm)

Figure 5

Damping coefficients versus excitation frequency (centered rotor) (L/D = 1, C/R = 1.1·10−3 , Ω = 25 krpm)

Figure 6

A large displacement is a succession of small perturbation

Figure 7

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

Figure 8

Fixed and rotating reference frame

Figure 9

(a) Coefficients A0 of the second order transfer function approximation, (b) coefficients A1 of the second order transfer function approximation, (c) coefficients A2 of the second order transfer function approximation, (d) coefficients B0 of the second order transfer function approximation and (e) coefficients B1 of the second order transfer function approximation.

Figure 10

Static forces

Figure 11

Figure 12

Case a: Fluid Forces in X direction (orbit method)

Figure 13

Case a: Fluid forces in Y direction (orbit method)

Figure 14

Figure 15

Figure 16

Power spectrum of displacement in the X direction

Figure 17

Power spectrum of displacement in the Y direction

Figure 18

Fluid forces due to unbalance in the X direction (10 last periods)

Figure 19

Fluid forces due to unbalance in the Y direction (10 last periods)

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