The oil pressure law in journal bearings due to of the shaft conical motion.

Parausanu, Ioan ; Petre, Cristian ; Gheorghiu, Horia 等

1. INTRODUCTION

It is well known that journal bearings have high load capacity, low friction, and minimal wear and, therefore, they appear to be ideal candidates to replace roller bearings as rotor support elements in high speed turbo machinery.

Taking into account both hydrodynamic and squeeze effects, they could also be safety used as dampers in some aircraft engine applications or in high speed turbo machinery handling cryogenic liquids.. ( Baskharone et al., 1991)

The object of this paper is to develop a theoretical model for the problem of an unaligned journal bearing, considering first the stationary case, and then the squeeze effects, induced by the angular velocities of the shaft.

In the end of this work the pressure distribution law is obtained into an explicit form for both, stationary and unstationary cases.

2. NOMENCLATURE

B--width of bearing;

[bar.B]--dimensionless width of bearing, B/D;

d--diameter of journal;

D--diameter of bearing;

e--journal lateral eccentricity;

h--film thickness;

[bar.h]--dimensionless film thickness, 2h/J;

J--diametral clearance, D--d;

n--shaft speed;

x, y, z--cartesian co-ordinates (fixed xOyz--frame);

[x.sub.b], [y.sub.b]--co-ordinates of point Mb on the bearing contour;

[x.sub.s], [y.sub.s]--co-ordinates of point Ms on the shaft contour;

[bar.z]--dimensionless axial co-ordinate, 2z/B;

X,Y,Z- cartesian co-ordinates (rotated XOYZ-frame);

[alpha],[beta]- angular displacements;

[??,[??]--angular velocities;

[[??].sup.*], [[??].sup.*]--dimensionless angular velocities, ([??], [??])/([pi] * n);

[epsilon]--eccentricity ratio, 2e/J;

[PSI]--dimensionless clearance, J/D;

[eta]--dynamic viscosity;

[theta]--angular co-ordinate.

3. THE MATHEMATICAL MODEL

The narrow journal bearing with shaft angular displacements is shown in Figure 1. (Someya,1989) The following assumptions are made:

1. The lubricant is a Newtonian liquid of constant viscosity (isothermal flow).

2. The surfaces of the journal and the sleeve are rigid.

3. The order of angular displacements is given by the ratio between diametral clearance and the width of the bearing: [alpha], [beta] [approximately equal to] / [bar.B].

4. Journal and the bush are not aligned (the conical motion of the shaft is present, see Figure 1). (San Andres, 1992)

Taking a fixed initial cartesian frame-xOyz, with point O placed at the bearing central plane and Oz being the shaft axis in the aligned case, the angular tridimensional position of the unaligned journal can be described in a XOYZ-cartesian frame with the same origin but axes rotated with Euler's angles: [alpha] and [beta]. (San Andres, 1993)

The connection between these frames is given by two rotation matrices:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 1 OMITTED]

In the XOYZ-frame, the equation describes the journal contour is:

[X.sup.2] + [Y.sup.2] = [(d/2).sup.2] (2)

and taking into account both relation (1) and the assumption that angles [alpha] and [beta] are considered small perturbations, the xOyz-shaft equation becomes:

[(x + y[alpha][beta]- z[alpha]).sup.2] +[(y + z[beta]).sup.2] =[(d/2).sup.2] (3)

Besides, xOyz-bearing equation is:

[(x-e).sup.2] + [y.sup.2] = [(D/2).sup.2] (4)

where e is the standard lateral eccentricity. Now considering that y, [alpha] * [beta] in equation (3) as a second order term which can be neglected, and introducing an angular [theta] co-ordinate (measured from the fixed Ox-axis in the journal sense of rotation) the position of points [M.sub.b] and [M.sub.s] (on the bearing, respectively on the shaft contour at the same journal radius) is described by: (Suciu & Parausanu, 1996)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Thus, the oil film thickness can be expressed as:

h = [square root of [([x.sub.b] -[x.sub.s]).sup.2] + [([y.sub.b] - [y.sub.s]).sup.2]] (6)

Introducing equations (5) in (6) and neglecting under square root the term [[e sin [theta]-z ([alpha] sin [theta] + [beta] cos [theta])].sup.2], having the same order with the term [[e sin [theta]].sup.2] which is usually neglected in the hydrodynamic journal bearing theory, the film thickness is given by:

h [??] J/2 + e cos [theta] - z[alpha] cos [theta] + z[beta] sin [theta] (7)

or in dimensionless form:

[bar.h] = 2h/J =1 + [epsilon] cos [theta] [bar.B]/[PSI] [bar.z][alpha] cos [theta] + [bar.B]/[PSI] [bar.z][beta] sin [theta] (8)

where: [bar.z] = 2 * z/B; [bar.B] = B/D and [PSI] = J/D . The pressure is obtained by solving the Reynolds equation which for the unaligned narrow journal bearing has the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where the first term in the right member gives the squeeze film effects induced by the angular velocities [??] and [??]. When equation (9) is integrated twice along the axial co-ordinate z under boundary conditions:

p = 0 for z = [+ or -]B/2 (10)

the pressure distribution has the form:

p = 6[pi] * n [eta][[bar.B].sup.2]/[[PSI].sup.2] [E.sub.1] + 3[pi]* n [eta] [E.sub.2]([E.sub.3] - [E.sub.4]) (11)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

and

K = [bar.B]/[PSI] * [beta] sin [theta] - [alpha]cos [theta]/1 + [epsilon] cos [theta] (16)

4. CONCLUSION

The final aim of the authors is to improve the calculation of the critical rotor speed by including the flexibility of the bearing oil film due to the angular displacements of the shaft.

So, this analysis occurs as a complete preliminary hydrodynamic step, which will be followed by another rotor-dynamic step.

In this work the pressure distribution is set into an explicit form for both, stationary and non-steady cases, the "pressure mountain" becoming strongly non-symmetric.

This work is just a preliminary announcement of some results regarding unaligned journal bearings.

5. REFERENCES

Baskharone, E.A. & Hensel, S.J. (1991). Interrelated Rotordynamic Effects of Cylindrical and Conical Whirl of Annular Seal Rotors, ASME, Journal of Tribology, Vol. 113, pp. 470-480.

San Andres, L.A. (1992). Approximate Analysis of Turbulent Hibrid Bearings: Static and Dynamic Performance for Centered Operations, ASME, Journal of Tribology, Vol. 112, pp. 692-698.

San Andres, L.A. (1993). Analysis of Hydrostatic Journal Bearings with End Seals, ASME, Journal of Tribology, Vol. 114, pp. 802-811.

Someya, T. (1989). Journal Bearing Databook, Springer Verlag, Berlin.

Suciu, C.V. & Parausanu, I. (1996). A Complet Investigation of unaligned narrow Journal Bearing, Mecanique appliquee, Vol. 41, pp. 99-120.