Determination of Dynamic Coefficients of Air-Ring Bearings

Abstract

Purpose

An externally-pressurized journal air bearing (AB) with an air-ring (AR), or air-ring bearing (ARB), for a balanced, rigid and light-weight rotor is studied. An elastic structure in the form of an AR is provided between the bearing-bushing and the casing. The ARB is analyzed to determine the dynamic coefficients (DC) at various angular velocities and angular frequencies of vibration of the journal in its range of operation. These DC can then be used to predict the dynamic stability of the rotor ARB system against self-excited (SE) vibration. A numerical simulation procedure is followed to determine the DC.

Methods

The ARB is modeled as a two-degrees of freedom system. During the simulation, the journal follows a prescribed harmonic motion. Self-exciting forces due to flow dynamics inside an ARB induce this motion. Three-dimensional (3-D) flow equations are solved on a moving/deformable grid using ANSYS®, to compute the pressure (p) distribution in the ARB. Unlike in previous studies, in this study the bushing displacement is determined by the instantaneous p-distribution in the ARB. DC of both AB and AR are determined simultaneously by considering the interaction between the AR and the AB regions through the feed-holes in the bushing.

Results

Time-dependent displacement, velocity, and load-carrying capacity obtained by numerical simulation are used to evaluate the DC.

Conclusion

Incorporation of an AR around an AB can prevent SE vibration due to positive values of direct damping coefficients of AR. A 3-D flow analysis can reveal the realistic nature of flow in an ARB.

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Abbreviations

AB:

Air-bearing

AR:

Air-ring

ARB:

Air-ring bearing

CG:

Geometric center

CFD:

Computational fluid dynamics

DC:

Dynamic coefficients

FC:

Face centroid

FH:

Feed-hole

GCI:

Grid convergence indicator

LHS:

Left-hand side

RARBS:

Rotor air-ring bearing system

RE:

Reynolds equation

RHS:

Right-hand side

RPM:

Revolutions per minute

SE:

Self-excited

SEP:

Static equilibrium position

SHM:

Simple harmonic motion

SSS:

Steady-state simulation

TSS:

Transient-state simulation

UDF:

User-defined function

1-DOF:

Single-degree of freedom

2-DOF:

Two-degrees of freedom

A :

Amplitude of vibration (m)

a :

A coefficient (see Eq. 5)

b :

Reference cell size (m); time step size (s); a coefficient (see Eq. 6)

C :

Damping coefficient (N s m1)

c :

Radial clearance (m); specific heat (J kg1 K1)

D :

Diameter (m)

d :

Diameter (m)

E :

Discretization error

e :

Eccentricity (m)

F :

Absolute dynamic load-carrying capacity (N)

f :

Incremental dynamic load-carrying capacity (N)

g :

Acceleration due to gravity (m s2)

h :

Axial length (m)

K :

Stiffness coefficient (N m1)

k :

Thermal conductivity (W m1 K1)

L :

Bearing length (m)

m :

Mass (kg)

N :

Number of time steps; rotational speed of the rotor (RPM)

n :

Number of feed-holes per row (see Table 1); an index

P :

Static load (N)

p :

Static pressure (Pa)

R :

Gas constant (J kg1 K1)

r :

Radial coordinate (m); number of rows of feed-holes (see Table 1)

S :

Absolute displacement (m)

\(\dot{S}\) :

Absolute velocity (m s1)

s :

Incremental displacement (m)

\(\dot{s}\) :

Incremental velocity (m s1)

T :

Temperature (K); time period of vibration (s)

t :

Time (s)

U :

A target quantity of interest

u :

Fluid velocity component (m s1)

x :

A coordinate (m)

y :

A coordinate (m)

z :

A coordinate (m)

α :

Attitude angle (rad)

β :

Angular position of baffle (rad) (see Fig. 1)

γ :

Angular width of baffle (rad) (see Fig. 1)

δ :

Radial distance (across film thickness) measured from bushing surface (m)

\(\epsilon\) :

(= e/c) Eccentricity ratio

θ :

Angle coordinate (rad)

λ :

Rate of reduction of truncation error

μ :

Dynamic viscosity (kg m1 s1)

ν :

Angular frequency (rad s1)

ρ :

Density (kg m3)

ω :

Angular velocity (rad s1)

AB:

Air-bearing

AR:

Air-ring

B:

Bushing

dm:

Dynamic mesh

i :

Grid-level

J:

Journal

max:

Maximum

v :

At constant volume

r :

Radial

rel:

Relative

s:

Supply

z :

Axial

θ :

Circumferential

i :

An index

j :

An index

x :

x-Component

y :

y-Component

F :

Absolute dynamic load-carrying capacity (N)

f :

Incremental dynamic load-carrying capacity (N)

r :

Position vector (m)

S :

Absolute displacement (m)

\(\dot{\mathbf{S}}\) :

Absolute velocity (m s1)

s :

Incremental displacement (m)

\(\dot{\mathbf{s}}\) :

Incremental velocity (m s1)

\(\ddot{\mathbf{s}}\) :

Acceleration (m s2)

u :

Velocity (m s1); orthonormal vector

v :

Orthogonal vector

w :

A vector

τ :

Viscous stress tensor (N m2)

ω :

Angular velocity vector (rad s1)

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Acknowledgements

The corresponding author wishes to thank J-U. Friemann, X. Wang and P. Zamankhan, of ANSYS® Inc. for their guidance in writing UDFs for moving/deformable mesh in the transient-state simulations using ANSYS® FLUENT.

Funding

This research did not receive any grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Correspondence to Muruganandam Muthanandam.

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Muthanandam, M., Thyageswaran, S. Determination of Dynamic Coefficients of Air-Ring Bearings. J. Vib. Eng. Technol. 9, 1–21 (2021). https://doi.org/10.1007/s42417-020-00207-4

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Keywords

  • Journal bearings
  • Externally-pressurized air bearings
  • Dynamic coefficients
  • Self-excited whirl
  • Steady-state simulation
  • Transient-state simulation