Handbook of Lubrication Episode 1 Part 6 ppt

In thrust bearings somewhat more pocket area has historically been used than in journal bearings, about 5% of the bearing area.. PLANAR BEARINGS IN NORMAL APPROACH For isoviscous planar

FIGURE 12 Two elliptical oil lift pockets located at quarter points along a journal bearing axis Removing the plug allows the supply passages to be flushed.

with size AL, the area of a single bearing pocket, is usually made about 1.5% of A in a

bearing of this type

Once the load is lifted, the pocket pressure falls because pressurized oil is distributed over the whole bearing area The flow, oil film thickness, and other quantities can be calculated for this condition as a pure hydrostatic bearing (before rotation starts) The lift

is the following function of the purnp flow:

(14)

This is the corrected form of Equation (12–19) in Reference 12

A numerical example will illustrate the use of the above equations, where Q = 0.07 /

sec (1.1 gpm), D = 533.4 mm (21 in.), L = 304.8 mm (12 in.), CD = 0.711 mm (0.028 in.), W = 382,500 N (86,000 lb), n = 2 pockets, AL = 2580 mm2 (4 in.2), and μ = 0.058 Pa·sec (8.4 µreyn)

Solving Equation 13, assuming KBA = 3, gives a breakaway pressure of 21.1 MPa (3072 psi) The oil pump should be sized to give at least this much pressure with some margin (say, 50 to 100%) to allow for performance deterioration over a period of time Employing appropriate units, Equation 14 gives an eccentricity ratio of 0.63 Assuming that the shaft moves straight up, the lift equals the minimum film thickness, (1 – )(CD/2), or 0.132 mm

(0.005 in.) This is a realistic value which will allow for some misalignment or shaft deflection

In thrust bearings somewhat more pocket area has historically been used than in journal bearings, about 5% of the bearing area The pad in Figure 13 is from a bearing with an outside diameter of 2286 mm (90 in.) with a design load of 4.1 MPa (600 psi) At breakaway the pocket pressure rises to 12.4 MPa (1800 psi) It then falls back to a steady-state value

of 4.8 MPa (700 psi)

‘The lift (h) may be estimated by assuming a circular pressure distribution similar to the bearing in Figure 3 For the bearing pad in Figure 13 the following values may be assigned

to the variables: Q = 0.025 /sec (0.4 gpm), Pp = 4.8 MPa (700 psi), μ = 0.056 Pa·sec (56 cP), (ISO VG 68 oil at 38°C), R = 286 mm (11.3 in.), and Ro= 51 mm (2.0 in.)

PBA = Breakaway pressure in oil lift

Po = Ambient pressure around bearing sealing land

Ps = Pressure in bearing pocket

Ps = Supply pressure

Q = Flow

Qc = Capillary tube flow

Qk = Constant flow

Qo = Orifice flow

Q– = Flow rate parameter, Reference 3

R = Outside radius of thrust bearing (Figure 3)

Ro = Pocket radius of thrust bearing (Figure 3)

w = Width of land normal to direction of flow

W = Load

 = Eccentricity ratio = e  radial clearance

ρ = Fluid density

θ = Angle subtended by circumferential land in journal bearing (Figure 10)

φ = Direction of loading in journal bearing (zero is toward a pocket) (Figure 10)

μ = Fluid viscosity

REFERENCES

1 Anon., Floating shoes form big bearings, Mach Design, 49(27), 37, 1977.

2 Rippel, H C., Cast Bronze Hydrostatic Bearing Design Manual, Cast Bronze Bearing Institute, Chicago,

1975.

3 Stout, K J and Rowe, W B., Externally pressurized bearings — design for manufacture III Design

of liquid externally pressurized bearings for manufacture including tolerancing procedures, Tribol Int.,

7(5), 195, October 1974.

4 Rowe, W B., O’Donoghue, J P., and Cameron, A., Optimization of externally pressurized bearings

for minimum power and low temperature rise, Tribology, 3(4), 153, August 1970.

5 Sneck, H J., A survey of gas-lubricated porous bearings, Trans ASME, Ser F, 90(4), 804, October 1968.

6 Fuller, D D., Hydrostatic lubrication, in Standard Handbook of Lubrication Engineering, O’Connor, J.

J., Boyd, J., and Avallone, E A., Eds., McGraw-Hill, New York, 1968, 3-17.

7 Szeri, A Z., Hydrostatic bearings, in Tribology: Friction, Lubrication, and Wear, Szeri, A Z., Ed.,

McGraw-Hill, New York, 1980, 47.

8 Elwell, R C and Sternlicht, B., Theoretical and experimental analysis of hydrostatic thrust bearings,

Trans ASME, Ser D, 82(3), 505, September 1960.

9 Raimondi, A A and Boyd, J., Hydrostatic journal bearings (compensated), in Standard Handbook of

Lubrication, O’Connor, J J., Boyd, J., and Avallone, E A., Eds., McGraw-Hill, New York, 1968,

5-66.

10 Hunt, J B and Ahmed, K M., Load capacity, stiffness and flow characteristics of a hydrostatically

lubricated six-pocket journal bearing supporting a rotary spindle, Part 3N, Proc I.M.E., 182, 53, 1967-8.

11 O’Donoghue, J P and Rowe, W B., Hydrostatic bearing design, Tribology, 2(1), 25, February 1969.

12 Wikock, D F and Booser, E R., Bearing Design and Application, McGraw-Hill, New York, 1957.

SQUEEZE FILMS AND BEARING DYNAMICS

J F Booker

INTRODUCTION

This chapter covers transient behavior of viscous lubricant films under loads which may

be fixed or variable in magnitude and/or direction Since it takes time for such films to be

squeezed out from between surfaces, bearings can often carry surprisingly high peak loads

as compared to those they might sustain in steady-state operation

“Squeeze-film” action is often of interest because of the damping it provides Occasionally such special devices as dampers for turbomachinery are involved; more often, as in recip-rocating machinery, the damping action is provided by conventional bearings

The following analysis begins with treatment of the normal approach of planar bearings

It proceeds with examination of cylindrical bearings in one- and two-dimensional translation, both without and with accompanying rotation Finally, by way of an example for connecting-rod bearings, analysis is supplemented by a parametric design study and correlation of a failure criterion with field experience

GENERAL REYNOLDS EQUATION

In its general form the incompressible Reynolds equation derived in an earlier chapter can be written in rectangular coordinates x, y

or in polar coordinates, r, θ

For an important class of normal approach “squeeze film” problems, the average tan-gential surface velocity U— has negligible effect, leaving only the squeeze rate ∂h/∂t as an effective driving term

PLANAR BEARINGS IN NORMAL APPROACH

For isoviscous planar normal approach with uniform film thickness, the Reynolds equation simplifies in rectangular coordinates to

or in polar coordinates

Special Formulation for Circular Section

As an example, consider Figure 1 in which a film is squeezed by the normal approach

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where, for a constant load

For particular fixed values µ, R, and F, the above relations give an approach rate slowing asymptotically as final closure is approached This qualitative behavior is typical of all

“squeeze films” in response to time integral (impulse), not instantaneous values of loading.

General Formulation

The relations derived for the circular section are also valid for general geometries if expressed in terms of area A and dimensionless shape factors P and K Thus,

Shape factor P is a measure of the sharpness or nonuniformity of the pressure distribution,

K the dynamic stiffness or damping rate of the lubricant film as a whole

Circular Section

The circular section in the example has area A = πR2and shape factors

P = 2 and K = —3

2π = 0.477

Elliptical Section

An elliptical section with major and minor diameters L and B has area A = πLB/4 and shape factors as shown in Figure 2

P = 2 and 1/K = (B/L + L/B) π /3

Note the reduction to the circular section result as slenderness ratio B/L→ 1

Rectangular Section

A rectangular section with sides L and B has area A = LB and shape factors P and K

as shown in Figure 3 Though these results have been computed from an exact series solution,1 they are quite accurately fit by the optimum approximate Warner solution2,3expressions

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“Narrow-Section” Formulas

The previous results for rectangular sections show the asymptotic behavior

K → L/B and P → 3/2 while holding

A = LB as B/L → 0 These relations, which correspond to a one-dimensional parabolic pressure distribution (usually attributed to Sommerfeld), are applicable to any narrow section For example, they hold in the limit for an annular ring with relatively similar inner and outer radii, corresponding

to many simple thrust bearings

“Broad-Section” Formulas

The previous results for elliptical sections can be expressed as

in terms of area and polar moment

These relations, which hold exactly for elliptical (and circular) sections, are also applicable approximately to any broad section

Application of this approximation (usually attributed to Saint Venant) to the rectangular

section studied previously gives

P≈ 2 and I/K ≈ (B/L + L/B) (π/3)2

so that for a square section

P≈ 2 and K ≈ (1/2) (3/π)2 = 0.456

as compared to the approximate values computed from the Warner solution above

and the numerically exact series values plotted in Figure 3

Similarly, application of the approximations to an equilateral triangular section gives

as compared to exact values

P = 20/9 = 2.222 and K = √–3/5 = 0.346

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Other Sections and Surfaces

Though the literature1,4-6 contains exact formulas for normal approach of many other special planar sections (including complete and annular circles and sectors), the results given here should be entirely adequate for most purposes The literature1,4-8 also contains results for normal approach of a variety of nonplanar surfaces, including plates with small curvature (single and double), cones (complete and truncated), and spheres of various extents

CYLINDRICAL JOURNAL BEARINGS9-13

The “squeeze-film” behavior of nonrotating cylindrical bearings in one-dimensional radial

motion is qualitatively quite similar to that for planar bearings in normal approach, and

generalization to two-dimensional motion is conceptually straightforward Remarkably, even the addition of journal rotation causes no real difficulties Thus solution of general cylindrical journal bearing dynamics problems rests on an understanding of “squeeze-film” behavior

in simpler nonrotating cases

One-Dimensional Motion Without Rotation

Figure 4 shows a nonrotating journal moving radially downward into a cylindrical half-sleeve As before, rigorous analysis proceeds from the general Reynolds equation in rec-tangular coordinates wrapped around the journal circumference, a procedure justified by the clearance ratio h/R << 1 Tangential surface velocities are neglected Fully flooded ambient boundary conditions assumed at the axial and circumferential ends of the bearing film complete specification of the problem

Solution for pressure, etc., can be numerical or semianalytical.11In the latter case, com-putations are facilitated by special tables23for the “journal bearing integrals” which arise

General Formulation

Relations analogous to previous ones can be expressed in terms of dimensional geometrical

and material factors µ, L, D, R, and C and dimensionless functions P, Q, W, M, and J of

dimensionless slenderness ratio L/D and dimensionless eccentricity ratio < 1 (Recall that

previous dimensionless quantities for planar bearings were constants.)

Thus,

∋

126 CRC Handbook of Lubrication

and

For liquid films, which will not support significant negative pressures without rupturing,

the short-bearing results given here for the half-sleeve bearing of Figure 4 apply equally well to radial motion of the full-sleeve bearing of Figure 6.

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FIGURE 5 Characteristics for cylindrical bearings in one-dimensional motion (short-bearing film model) (a) Pressure vs eccentricity, (b) impedance and mobility vs eccentricity, and (c) impulse vs eccentricity.

vector or scalar quantities plotted over the clearance space of all possible eccentricity ratios Figure 8 allows a comparison of typical maps9-13for the liquid-film short-bearing model (in which film pressure is positive throughout the bearing half with normally approaching surfaces and vanishes in the other) The maps are oriented to velocity or force directions as shown Dashed/solid curvilinear families indicate magnitude/direction of pressure ratio, mobility, and impedance vectors in Figures 8a, b, and c, respectively Though the same basic data are displayed in both impedance and mobility maps, each point on one map corresponds to a (different) point on the other In particular, the sample points indicated in Figures 8b and c do not correspond

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FIGURE 7 Coordinate axes and vectors for two-dimensional motion.

FIGURE 8 Characteristics for cylindrical bearing in two-dimensional motion (short-bearing film model) 9-3 (a) Pressure vs eccentricity, (b) mobility vs eccentricity, and (c) impedance vs eccentricity.

Generally, such maps are specific to a particular slenderness ratio; for the short-bearing

film model vector P is entirely independent of ratio L/D, while vector W (or M) varies with

its square

One-dimensional Figures 5a and b correspond to the midlines of two-dimensional Figures 8a, , and c Similarly, the two-dimensional short-bearing approximations

are generalizations of the one-dimensional approximations given earlier More exact map data are available else where.9-13,24,25

Application of the map data to nonrotating bearings is straightforward: specification of e and e . allows direct determination of F via W; specification of e and F allows direct deter- mination of p* (or e . ) via P (or M) Transformations are required if (as is often the case)

the map frames x, y and/or x′,y′ do not coincide instantaneously with the computation frame X,

Y (Graphically, this simply requires rotating maps.)

General Formulation — With Rotation

For extension of these procedures to problems involving rotation of journal and/or sleeve, consider an “observer” fixed to the sleeve center but rotating at the average angular velocity

ω− of journal and sleeve (positive CCW) The absolute journal center velocity e.abs seen in

the “fixed” computation frame, X, Y and the relative velocity e .rel apparent to the observer

are related to journal eccentricity e and ω− by the simple kinematic expression

Since the average angular velocity of journal and sleeve (fluid entrainment velocity) apparent

to this observer would vanish, maximum pressure p* and resultant force F would seem to

be related solely to the relative (squeeze) velocity e .rel in exactly the same way as for the nonrotating bearings considered previously.

Thus extension of the previous procedures to general problems requires only use of the

kinematic relation above before the impedance procedure for finding force from (relative) velocity and/or after the mobility procedure for finding (relative) velocity from force; the

procedure for finding maximum pressure from force requires no modification, however The impedance and mobility methods are perfect complements Both provide for efficient

storage of bearing characteristics based on any suitable film model Because pressure

dis-tributions are not calculated, both methods permit efficient computation In appropriate applications the resulting equations of motion are in explicit form, and iterative calculations can thus be avoided in most system simulation studies

Since the impedance formulation is appropriate to cases in which instantaneous force is

desired, it seems most suited to problems in rotating machinery, particularly with damper

bearings

Since the mobility formulation is appropriate when instantaneous force is known, it has

found widest application in reciprocating machinery By giving instantaneous journal center velocity, the mobility method provides a basis for predicting an entire journal center path

by numerical extrapolation (while allowing simultaneous prediction of maximum film pres-sure) Numerical implementation of the mobility method is straightforward; simplified ver-sions require as few as 50 steps on programmable calculators A digital computer program which accepts tabulated duty cycle data can be compiled from about 200 FORTRAN statements.12,13

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