30079_21b.pdf

Table 21.4 Pressure-Viscosity Coefficients for Test Fluids at Three Temperatures

(From Ref. 17)

Temperature, 0 C

38 99 149

Pressure-viscosity Coefficient, f, m 2 /N

1.28 X 10~ 8

Test Fluid

Advanced ester

Formulated advanced ester

Polyalkyl aromatic

Polyalkyl aromatic + 10 wt % heavy resin

Synthetic paraffinic oil (lot 3)

Synthetic paraffinic oil (lot 4)

Synthetic paraffinic oil (lot 4) + antiwear additive

Synthetic paraffinic oil (lot 2) + antiwear additive

C-ether

Superrefined naphthenic mineral oil

Synthetic hydrocarbon (traction fluid)

Fluorinated polyether

0.851 X IO" 8

0.987 X 10~ 8

1.37

1.00

.874

1.58

1.25

1.01

1.70

1.28

1.06

1.77

1.51

1.09

1.99

1.51

1.29

1.96

1.55

1.25

1.81

1.37

1.13

1.80

.980

.795

2.51

1.54

1.27

3.12

1.71

.939

4.17

3.24

3.02

(2U1 )

^>° ^JI [(,-W-W"

2 f f

p(x,y) dx.dy,

where

/1 - v\ 1 - I/A' 1

E' = 2 ——+ ——

(21.12)

^b I

and v = Poisson's ratio

E = modulus of elasticity, N/m 2

Therefore, Eq. (21.6) is normally involved in hydrodynamic lubrication situations, while Eqs.

(21.7)-(21.11) are normally involved in elastohydrodynamic lubrication situations.

21.2 HYDRODYNAMIC AND HYDROSTATIC LUBRICATION

Surfaces lubricated hydrodynamically are normally conformal as pointed out in Section 21.1.1. The

conformal nature of the surfaces can take its form either as a thrust bearing or as a journal bearing,

both of which will be considered in this section. Three features must exist for hydrodynamic lubri-

cation to occur:

1. A viscous fluid must separate the lubricated surfaces.

2. There must be relative motion between the surfaces.

3. The geometry of the film shape must be larger in the inlet than at the outlet so that a

convergent wedge of lubricant is formed.

If feature 2 is absent, lubrication can still be achieved by establishing relative motion between the

fluid and the surfaces through external pressurization. This is discussed further in Section 21.2.3.

In hydrodynamic lubrication the entire friction arises from the shearing of the lubricant film so

that it is determined by the viscosity of the oil: the thinner (or less viscous) the oil, the lower the

friction. The great advantages of hydrodynamic lubrication are that the friction can be very low

(IJL =* 0.001) and, in the ideal case, there is no wear of the moving parts. The main problems in

hydrodynamic lubrication are associated with starting or stopping since the oil film thickness theo-

retically is zero when the speed is zero.

The emphasis in this section is on hydrodynamic and hydrostatic lubrication. This section is not

intended to be all inclusive but rather to typify the situations existing in hydrodynamic and hydrostatic

lubrication. For additional information the reader is recommended to investigate Gross et al., 19

Reiger, 2 0 Pinkus and Sternlicht, 2 1 and Rippel. 2 2

21.2.1 Liquid-Lubricated Hydrodynamic Journal Bearings

Journal bearings, as shown in Fig. 21.8, are used to support shafts and to carry radial loads with

minimum power loss and minimum wear. The bearing can be represented by a plain cylindrical bush

wrapped around the shaft, but practical bearings can adopt a variety of forms. The lubricant is supplied

at some convenient point through a hole or a groove. If the bearing extends around the full 360° of

the shaft, the bearing is described as a full journal bearing. If the angle of wrap is less than 360°,

the term "partial journal bearing" is employed.

Plain

Journal bearings rely on the motion of the shaft to generate the load-supporting pressures in the

lubricant film. The shaft does not normally run concentric with the bearing center. The distance

between the shaft center and the bearing center is known as the eccentricity. This eccentric position

within the bearing clearance is influenced by the load that it carries. The amount of eccentricity

adjusts itself until the load is balanced by the pressure generated in the converging portion of the

bearing. The pressure generated, and therefore the load capacity of the bearing, depends on the shaft

eccentricity e, the frequency of rotation N 9 and the effective viscosity of the lubricant 77 in the

converging film, as well as the bearing dimensions / and d and the clearance c. The three dimen-

sionless groupings normally used for journal bearings are:

1. The eccentricity ratio, e = etc

2. The length-to-diameter ratio, A = Ud

3. The Sommerfeld number, Sm = r)Nd 3 l/2Fc 2

When designing a journal bearing, the first requirement to be met is that it should operate with

an adequate minimum film thickness, which is directly related to the eccentricity (h min = c — e}.

Figures 21.9, 21.10, and 21.11 show the eccentricity ratio, the dimensionless minimum film thickness,

and the dimensionless Sommerfeld number for, respectively, a full journal bearing and partial journal

bearings of 180° and 120°. In these figures a recommended operating eccentricity ratio is indicated

as well as a preferred operational area. The left boundary of the shaded zone defines the optimum

eccentricity ratio for minimum coefficient of friction, and the right boundary is the optimum eccen-

tricity ratio for maximum load. In these figures it can be observed that the shaded area is significantly

reduced for the partial bearings as compared with the full journal bearing. These plots were adapted

from results given in Raimondi and Boyd. 2 3

Figures 21.12, 21.13, and 21.14 show a plot of attitude angle $ (angle between the direction of

the load and a line drawn through the centers of the bearing and the journal) and the bearing char-

acteristic number for various length-to-diameter ratios for, respectively, a full journal bearing and

partial journal bearings of 180° and 120°. This angle establishes where the minimum and maximum

film thicknesses are located within the bearing. These plots were also adapted from results given in

Raimondi and Boyd, 2 3 where additional information about the coefficient of friction, the flow variable,

the temperature rise, and the maximum film pressure ratio for a complete range of length-to-diameter

ratios as well as for full or partial journal bearings can be found.

Fig. 21.8 Journal bearing.

Fig. 21.9 Design figure showing eccentricity ratio, dimensionless minimum film thickness, and

Sommerfeld number for full journal bearings. (Adapted from Ref. 23.)

Nonplain

As applications have demanded higher speeds, vibration problems due to critical speeds, imbalance,

and instability have created a need for journal bearing geometries other than plain journal bearings.

These geometries have various patterns of variable clearance so as to create pad film thicknesses that

have more strongly converging and diverging regions. Figure 21.15 shows elliptical, offset half, three-

lobe, and four-lobe bearings—bearings different from the plain journal bearing. An excellent discus-

sion of the performance of these bearings is provided in Allaire and Flack, 2 4 and some of their

conclusions are presented here. In Fig. 21.15, each pad is moved in toward the center of the bearing

some fraction of the pad clearance in order to make the fluid-film thickness more converging and

diverging than that which occurs in a plain journal bearing. The pad center of curvature is indicated

by a cross. Generally, these bearings give good suppression of instabilities in the system but can be

subject to subsynchronous vibration at high speeds. Accurate manufacturing of these bearings is not

always easy to obtain.

Fig. 21.10 Design figure showing eccentricity ratio, dimensionless minimum film thickness, and

Sommerfeld number for 180° partial journal bearings, centrally loaded. (Adapted from Ref. 23.)

Fig. 21.11 Design figure showing eccentricity ratio, dimensionless minimum film thickness, and

Sommerfeld number for 120° partial journal bearings, centrally loaded. (Adapted from Ref. 23.)

Fig. 21.12 Design figure showing attitude angle (position of minimum film thickness) and Som-

merfeld number for full journal bearings, centrally loaded. (Adapted from Ref. 23.)

Fig. 21.13 Design figure showing attitude angle (position of minimum film thickness) and Som-

merfeld number for 180° partial journal bearings, centrally loaded. (Adapted from Ref. 23.)

Fig. 21.14 Design figure showing attitude angle (position of minimum film thickness) and Som-

merfeld number for 120° partial journal bearings, centrally loaded. (Adapted from Ref. 23.)

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