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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 Volume II 121 Copyright © 1983 CRC Press LLC
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SQUEEZE FILMS AND BEARING DYNAMICS

J. F. Booker

INTRODUCTION

This chapter covers transient behavior of viscous lubricant films under loads which maybe fixed or variable in magnitude and/or direction. Since it takes time for such films to besqueezed out from between surfaces, bearings can often carry surprisingly high peak loadsas compared to those they might sustain in steady-state operation.

“Squeeze-film” action is often of interest because of the damping it provides. Occasionallysuch 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 afailure criterion with field experience.

GENERAL REYNOLDS EQUATION

In its general form the incompressible Reynolds equation derived in an earlier chaptercan 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 equationsimplifies in rectangular coordinates to

or in polar coordinates

Special Formulation for Circular SectionAs an example, consider Figure 1 in which a film is squeezed by the normal approach

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of a circular plate. Fully flooded boundary conditions at ambient pressure simply require pto vanish at radius R. These boundary conditions and the Reynolds equation give

with the maximum (central) value

Resultant volumetric outflow rate is then

while the normal force applied to the lubricant film by the moving plate is

Inverting gives the equation of motion

While numerical integration over time (e.g., by Euler’s linear extrapolation) is straight-forward, formal integration over some interval of approach gives the general solution

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FIGURE 1. Planar circular sectionin normal approach.

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For particular fixed values µ, R, and F, the above relations give an approach rate slowingasymptotically 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 FormulationThe 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 SectionThe circular section in the example has area A = πR2 and shape factors

P = 2 and K = 3—2π = 0.477

Elliptical SectionAn 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 SectionA 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 solution2,3 expressions

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where A2 = 5/2 and B/L≤ 1

which also show the exact asymptotic behavior of Figure 3 as slenderness ratio B/L → 0.

Approximate RelationsFigures 2 and 3 show a general insensitivity to slenderness ratio, which suggests wide

applicability of rough “rule-of-thumb” approximations

P ≈ 2 and K ≈ 1/2

Two somewhat more elegant approximations follow.

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FIGURE 2. Shape factors for elliptical section.

FIGURE 3. Shape factors for rectangular section.

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“Narrow-Section” FormulasThe 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, theyhold in the limit for an annular ring with relatively similar inner and outer radii, correspondingto many simple thrust bearings.

“Broad-Section” FormulasThe 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 applicableapproximately to any broad section.

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

P ≈ 2.167 and K ≈ 0.419

and the numerically exact series values plotted in Figure 3

P = 2.100 and K = 0.421

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 SurfacesThough 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 givenhere should be entirely adequate for most purposes. The literature1,4-8 also contains resultsfor 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 radialmotion is qualitatively quite similar to that for planar bearings in normal approach, andgeneralization to two-dimensional motion is conceptually straightforward. Remarkably, eventhe addition of journal rotation causes no real difficulties. Thus solution of general cylindricaljournal bearing dynamics problems rests on an understanding of “squeeze-film” behaviorin simpler nonrotating cases.

One-Dimensional Motion Without RotationFigure 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 theclearance ratio h/R << 1. Tangential surface velocities are neglected. Fully flooded ambientboundary conditions assumed at the axial and circumferential ends of the bearing filmcomplete specification of the problem.

Solution for pressure, etc., can be numerical or semianalytical.11 In the latter case, com-putations are facilitated by special tables23 for the “journal bearing integrals” which arise.

General FormulationRelations 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 ofdimensionless slenderness ratio L/D and dimensionless eccentricity ratio < 1. (Recall thatprevious dimensionless quantities for planar bearings were constants.)

Thus,

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Evidently,

while the last three factors are related by

so that

Pressure ratio9 P( ,L/D) is a measure of the sharpness or nonuniformity of the pressuredistribution; flow factor Q( ,L/D) the outward flow responsiveness to loading; impedance10

W( ,L/D) the stiffness or damping rate; mobility11-15 M( ,L/D) the velocity responsivenessto loading; and impulse15,16 J( ,L/D) the displacement responsiveness to loading over aninterval.

Short-Bearing RelationsWhile numerically exact solutions are available elsewhere,17,18 the results given here are

obtained by the qualitatively correct and widely used short-bearing approximation discussedin previous chapters. For this particular solution (with its parabolic axial pressure distri-bution), factors P and Q are entirely independent of ratio L/D, while factors J and W (orM) vary with its square. Figure 5 shows data from several sources9-11,19 for L/D = 1;adjustments are easily made for other slenderness ratios.

Though computed from closed-form expressions for the short-bearing model, these dataare quite accurately fit by the approximations

∋∋∋

∋∋

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FIGURE 4. Cylindrical bearingin one-dimensional motion.

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so

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 equallywell 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.

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Two-Dimensional MotionGeneral Formulation — Without Rotation

Figure 6 shows a non-rotating journal moving in an arbitrary direction within a cylindricalsleeve. Fully flooded conditions assumed at the axial ends of the film would appear tocomplete specification of the problem. For liquid films, however, special analytical arrange-ments must be made to avoid negative pressures in the bearing half with normally recedingsurfaces.

The journal motion in the clearance space is now two-dimensional. Since journal eccen-tricity, squeeze velocity, maximum film pressure, and resultant film force (applied by thejournal to the lubricant film) all have magnitude and direction, previous scalar relationsmust be replaced by the vector forms

so that

involving dimensionless eccentricity ratio, pressure ratio, force (impedance), and velocity(mobility) vectors �, P, W, and M. Scalar impulse relations for the two-dimensional problemand their relationship to mobility relations are discussed elsewhere.15,16

The various vectors can be displayed in “fixed” coordinates X, Y or in “moving”coordinates x, y or x′,y′ referenced, respectively, to velocity or force directions as shown inFigure 7. Using the two “moving” frames, bearing data can be displayed in maps of

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FIGURE 6. Cylindrical bearingin two-dimensional motion.

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

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FIGURE 7. Coordinate axes and vectorsfor 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.

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FIGURE 8b

FIGURE 8 c

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Generally, such maps are specific to a particular slenderness ratio; for the short-bearingfilm model vector P is entirely independent of ratio L/D, while vector W (or M) varies withits square.

One-dimensional Figures 5a and b correspond to the midlines of two-dimensional Figures8a, b, and c. Similarly, the two-dimensional short-bearing approximations

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

Application of the map data to nonrotating bearings is straightforward: specification of eand 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 RotationFor 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 velocityω− 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) apparentto this observer would vanish, maximum pressure p* and resultant force F would seem tobe 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; theprocedure for finding maximum pressure from force requires no modification, however.

The impedance and mobility methods are perfect complements. Both provide for efficientstorage of bearing characteristics based on any suitable film model. Because pressure dis-tributions are not calculated, both methods permit efficient computation. In appropriateapplications the resulting equations of motion are in explicit form, and iterative calculationscan thus be avoided in most system simulation studies.

Since the impedance formulation is appropriate to cases in which instantaneous force isdesired, it seems most suited to problems in rotating machinery, particularly with damperbearings.

Since the mobility formulation is appropriate when instantaneous force is known, it hasfound widest application in reciprocating machinery. By giving instantaneous journal centervelocity, the mobility method provides a basis for predicting an entire journal center pathby 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 programwhich accepts tabulated duty cycle data can be compiled from about 200 FORTRANstatements.12,13

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Connecting-Rod Bearing Example13

Orbit Computation by the Mobility MethodConnecting-rod bearings have complex loading as well as unsteady angular motion. En-

gine, bearing, and lubricant parameters for one such bearing are given elsewhere,12,15 togetherwith load components in a coordinate system X, Y fixed to the (moving) connecting rod.Figure 9a shows a full cycle of such loading; Figure 9b shows the corresponding steady-state displacement response computed using the short bearing model data represented inFigure 8b.

Computations using essentially the same program with data for more accurate film modelsshow qualitatively similar results.12,13,25,26 Extensive comparisons20 support continued useof the short-bearing model in correlation studies. Further applications of orbit analysis,parametric studies, corresponding design criteria (minimum film thickness, maximum filmpressure, power dissipation, etc.), and their correlation with failure modes and field expe-rience are discussed elsewhere.13,20-22,26,27

Parametric StudiesMobility method results for four-stroke engines (automotive Otto and Diesel, and industrial

Diesel) are summarized elsewhere.20 In all cases, firing loads have a minor (less than 20%)effect on predicted minimum film thicknesses so long as the maximum bearing load due to

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FIGURE 9. Connecting-rod bearing polar diagrams.12,13,15 (a) Journal loading cycle (four-stroke combustion),and (b) journal displacement cycle (short-bearing film model).

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firing alone is no more than about seven times the maximum bearing load due to inertiaalone. For a particular medium-speed Diesel this means that firing loads have negligibleeffect on predicted film thicknesses above about 300 rev/min.

Thus for many higher-speed engines, firing loads have very little effect on minimum filmthickness and can be reasonably ignored in preliminary design computations. Figure 1013,21

summarizes 120 different minimum film thickness predictions for connecting-rod bearings

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loaded by inertia forces alone. The omission of firing loads allows convenient characterizationof results by the dimensionless parameters shown. (For film models other than the short-bearing, both upper and lower graphs will be weakly dependent on slenderness ratio L/D.)Thus preliminary design guidance is available through Figure 10 without resort to computa-tion.

The intermediate value 0 given by the lower graph of Figure 10 can be interpreted asthe steady-state response to a steadily rotating inertia load applied to a steadily rotatingjournal in a nonrotating sleeve. The final value max given by the upper graph of Figure 10reflects the further effects of reciprocating inertia and engine geometry on extremes ofperiodic response.

Field ExperienceFilm thickness predictions and field experience for about 60 practical connecting-rod

bearings,20 together with experiences from several other sources,22 suggest the danger levelsin Table 1. Main bearings are understood to be a bit more tolerant in smaller sizes.

Noting the uncertainties involved, exceeding these values in no sense guarantees success;though believed to be representative, they are offered for information only. It is also inter-esting to compare these limiting values of predicted oil film thickness with peak-to-valleyestimates of surface finish.

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Table 1DANGER LEVELS FOR FILM THICKNESSES

PREDICTED BY SHORT BEARING FILM MODELFOR CONNECTING-ROD BEARINGS13

D(typical) h(dangerous)mm(in.) µm(µin.)

Automotive (Otto) 50(2) 1.0 (40)Automotive (Diesel) 75—100 (3—4) 1.75 (70)Industrial (Diesel) 250(10) 2.5 (100)

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NOMENCLATURE

A Area [L2]I Area polar moment [L4]L Length [L]B Breadth [L]D Diameter [L]R Radius [L]C Radial clearance [L]U Surface tangential velocity [LT–l

ω Surface angular velocity [T–1]µ Film viscosity [FL–2T]h Film thickness [L]P Film pressure [FL–2]P Dimensionless pressure ratio [ – ]q Outflow rate [L3T–1]Q Dimensionless outflow ratio [ – ]F Film force [F]K Dimensionless film force (stiffness) [ – ]W Dimensionless film force (impedance) [ – ]M Dimensionless velocity (mobility) [ – ]J Dimensionless impulse [ – ]e Eccentricity [L]

Dimensionless eccentricity ratio [ – ]r Crank radius (throw) [L]� Rod length [L]mrec Reciprocating mass [M]mrot Rotating mass [M]t Time [T]r, θ Coordinates [L,–]x,y Coordinates [L,L]x′,y′ Coordinates [L,L]X,Y Coordinates [L,L]· Time derivative [T–1]* Special value– Average value

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REFERENCES

1. Gross, W. A., Matsch, L. A., Castelli, V., Eshel, A., Vohr, J. H., and Wildmann, M., Fluid FilmLubrication, John Wiley & Sons, New York, 1980, chap. 8.

2. Warner, P. C., Static and dynamic properties of partial journal bearings, J. Basic Eng., 85, 247, 1963.3. Booker, J. F. and Rohde, S. M., Toward the Optimum Side Leakage Correction Factor, Research Rep.,

General Motors Research Laboratories, Warren. Mich., in press.4. Archibald, F. R., Squeeze films, in Standard Handbook of Lubrication Engineering, O’Connor, J. J.,

Ed., McGraw-Hill, New York, 1968, chap. 7.5. Moore, D. F., A review of squeeze films, Wear, 8, 245, 1965.6. Moore, D. F., Principles and Applications of Tribology, Pergamon Press, Oxford, 1975, 113.7. Hays, D. F., Squeeze films for rectangular plates, J. Basic Eng., 83, 579, 1961.8. Goenka, P. K. and Booker, J. F., Spherical bearings: stalic and dynamic analysis via the finite element

method, J. Lubr. Technol., 102, 308, 1980.9. Booker, J. F., Dynamically loaded journal bearings: maximum film pressure, J. Lubr. Technol., 91, 534,

1969.10. Childs, D., Moes, H., and van Leeuwen, H., Journal bearing impedance descriptions for rotordynamic

applications (with discussion by Booker, J. F.), J. Lubr. Technol., 99, 198, 1977.11. Booker, J. F., Dynamically loaded journal bearings: mobility method of solution, J. Basic Eng., 87, 537,

1965.12. Booker, J. F., Dynamically loaded journal bearings: numerical application of the mobility method. J. Lubr.

Technol., 93, 168 and 315, 1971.13. Booker, J. F., Design of dynamically loaded journal bearings, in Fundamentals of the Design of Fluid

Film Bearings, Rohde, S. M., Maday, C. J., and Allaire, P. E., Eds., American Society of MechanicalEngineers, New York, 1979, 31.

14. Barwell, F. T., Bearing Systems, Oxford University Press, Oxford, 1979, 261.15. Campbell, J., Love, P. P., Martin, F. A., and Rafique, S. O., Bearings for reciprocating machinery:

a review of the present slate of theoretical, experimental and service knowledge (with discussion by Booker,J. F.), Proc. Inst. Mech. Eng., 182(3A), 51, 1967.

16. Blok, H., Full journal bearings under dynamic duty: impulse method of solution and flapping action (withdiscussion by Booker, J. F.), J. Lubr. Technol., 97, 168, 1975.

17. Hays, D. F., Squeeze films: a finite journal bearing with a fluctuating load (with discussion by Phelan,R. M.), J. Basic Eng., 83, 579, 1961.

18. Donaldson, R. R., Minimum squeeze film thickness in a periodically loaded journal bearing, J. Lubr.Technol., 93, 130, 1971.

19. Booker, J. F. Analysis of Dynamically Loaded Journal Bearings: The Squeeze Film Considering Cavitation,Ph.D. thesis, Cornell University, Ithaca, N.Y., 1961.

20. New, N. H., The use of computer design techniques applied to IC engines, presented at Int. Symp. onPlain Bearings, S̀trbské Pleso, Czechoslovakia, October 24 to 26, 1972.

21. Martin, F. A. and Booker, J. F., Influence of engine inertia forces on minimum film thickness in con-rod big-end bearings, Proc. Inst. Mech. Eng., 181, 30, 1967.

22. Warriner, J. F., Factors affecting the design and operation of thin shell bearings for the modem dieselengine, Diesel Engines for the World, Whitehall Press, England, 1977/78, 13-23.

23. Booker, J. F., A table of the journal-bearing integral, J. Lubr. Technol., 87, 533, 1965.24. Moes, H. and Bosnia, R., Mobility and impedance definitions for plain journal bearings, J. Lubr. Technol.,

103, 468, 1981.25. Goenka, P. K., Analytical curve fits for solution parameters of dynamically loaded journal bearings, ASME

PaperNo. 83-Lub-33, J. Tribology, in press.26. Martin, F. A., Developments in engine bearings design, in Tribology International, 16, 147, 1983, from

Tribology of Reciprocating Engines, (Proc. 9thLeeds-Lyon Symp. on Tribology, Leeds, England, September1982). Dowson, D., Taylor, C. M., Godet, M., and Berth, D., Eds., Butterworths, 1983, 9.

27. Hollander, M. and Bryda, K. A., Interpretation of engine bearing performance by journal orbit analysis,Paper 830062, presented at SAE International Congress, Detroit, Mich., February 28 to March 4, 1983.

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