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TOApproved for public release, distributionunlimited

FROMDistribution authorized to U.S. Gov't.agencies and their contractors;Administrative/Operational Use; May 1965.Other requests shall be referred to theDirector, Air Force Aero PropulsionLaboratory, Attn: RTD, Wright-PattersonAFB, OH 45433.

AUTHORITY

AFAPL ltr, 21 Jul 1971

THIS PAGE IS UNCLASSIFIED

LLO

AFAPL -TR- 65-45Part IV

ROTOR-BEARING DYNAMICS DESIGN TECHNOLOGY,Part IV: Ball Bearing Design Data

P Lewis>" S L Malanoski

Mechanical Technology Incorporated

- •TECHNICAL REPORT AFAPL-TR-65-45,PART IV

may 1965 DODCi ~JUL2 196 W

Air Force Aem Propulsion Lahoatu'yResearch and Technology Division

Air Force System CommandWright-Patterson Air Force .Base, Okil

B

Best Available •Copy

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Copies of this report 4hould not be returnwe Lo the Research and Technology Div-

ision unless return is required by security considerations, contractual obligation.

or notice on a specific document. -...... ..NTION: Wem gmmuet or other Odzwfp, spee-ficatlons or other aak a% used far W• Punoeoother than in, canectim viLth & 1eflltj re,, ted.gove"raut P2ro at opaverati the U. B.0ovezmnt thezeby inec no reep;=ljj.ý-, noir aobligation *~ateoever; wd thes fact that tin Oow*=-Mnt My have ftnalate, faMiesh or In '&W Veysupplied the s aid dOwm s• spmifiostims, or otherdate is not to be reszd by ML11catlca or other.vise as In any er Ilom the bulder or MWother persm or corpo m or 0009 1.1% Uw rltwsor PemissIam to Nofmbim, use sel, M

PatetedInvention that mw in = W be elated.thereto.

LONAPI.T-54

AROTOR-BEARING DYNAMICS DESIGN TECHNOLOGY.Part WVe Ball1 Bearing esignl Data 1,ý

SBMalanoski. -

0 Mechanical Technology, Incepmdld., /V.,n,N. Y.*- Mi i , /TI- • -T/ -

S TECHNICAL PE .

0 may 165,

A F 4 i4..

Air Forco"e'uo Propulsion LaboratoryResearch and Technology Division

Air Force Systems CommandWright-Patterson Air Force Base, Ohio

FOREWOID

Thit report was prepared by Mechanical Technology Incorporated, 968 Albany-Shaker load, Latham, Hey York 12110, under USAF Contract No. AF 33(615)-1895.The contract was initiated under Project No. 3145, '"ynamic Energy ConversionTechnology", Task No. 314511, 'Nuclear Mechanical Power Units". The work wasadministered under the direction of the Air Force Aero Propulsion Laboratory,Research and Technology Division, with Mr. John L. Norris (APYL) acting asproject engineer. I

This report covers worl conducted from 1 April 1964 to I April 1965.

This report was submitted by the authors for review on 18 March 1965. It isPart IV of final documentation issued in multiple parts. This report alsois identified by the contractor's designation NTI-65-TI-35.

This technical report has been reviewed and is approved.

ARTHUR V. CHURCHILL, CheFuels and Lubricants BranchTechnical Support DivisionAir Force Aero Propulsion Laboratory

ABSTRACT

This Part IV of the Final Report presents design dsL& fr the stiffness character-

istics of ball bearings for use in analyzing thc dynamical perforPnance of a rotor.

'ae dynaic character!-tf_- of flutA f.1li bearings are given in PFor III which

also gives the methods for performing the analysis of the rotor-bearing system.

Design data are presented for the extra-light and light group of deep-grooved

and angular contact bearings undergoing either a pure radial load, puze axial

load, or combined radial load with axial preload. The data are given in graphical

form and cover both radial stiffness and loid-carrying capacity. A nominal damping

value for ball bearings, obtained from experimevtation, is suggested.

-, Some of the general guide rules for the selection of ball bearings are given.

These are concerned with fatigue life, limiting speeds, design, and lubrication.

Safe load levels are indicated.

Lit,

§

TABIZ O? COwTEWS

ABSTRA'i CT .................................... . . . ..... ... .... LLi

ILLUSTR•ATIONS ............... o...... #....................... v

1.• INTRODUCTION .......................... .......... 0.&. 0... t.... ...... 0

II. DESIGN REQUIREMENTS

Load Leve I ....................... ...........*• ........ 2

Ball bearing Dping.................•...................... 2

Race Curvatures ....... .......................... 2

Ball Bearings - Life............................o........... 4

Bearing Centrifugal Loadin ............. ..... 5

Effecte of Cyclic Loading on Bearin Lifen.................... 6

Lubricant Life ....... ... ............ 0.0..0.6..... . 0... ...... 12

III. DESIGN DATA

Description and Discuseion of Charts.........,............... 14

Table I Deep Groove Bearings ............. 0............. 17

Table I1 Angular Contact Barings...................... 18

Table III Axial Loaded Deep-Grooved Bearingso............ 19

Design Charts

Pure Radial Load ...................................... 20Pure Thrust Load .............. ,........ 25

Radial Load with Axial Preload......................0.... 34

IV. SAMPLE PROBLEMS TO ILLUSTRATE USE OF WSIGIF CHARTS

1. Pure Radial Loaded Bearing............................... 59

2. Unidirectional Thrust Loaded Bearinag..................... 59

3. Double-Acting Thrust Loaded Bearing ....................... 59

4. Radial Loaded, Axial Preloaded Angular Contact Bearing... 60

V. REFERENCES..... .... ............ ............ .62VI. APPENDIX

A. Analysis ........................................... 63

B. Computer Program, PN0182............................... 68

C. Nomenclature for Analysis................................ 76

iv

_L _ " .. .. 1-

ILLUSTRATIONS

FIGURE PAGE

A Ball Bearing Schematic ........................................... 3

B Equivalent Load Ratio as a Function of Cyclic Load Ratio ......... a

C Bearing Life Ratio as a Function of Equivalent Load Ratio 10

D Bearing Life versus Temperature .................................. 13

1 Radial Stiffness for Deep Gruove Ball Beari:g-Pure Radial Load... 21

2 Radial Stiffness for Deep Groove Ball Bearings-Pure Radial Load.. 22

3 Radial Stiffness for Deep Groove Ball Bearings-Pure Radial Load.. 23

4 Radial Stiffness for Deep Groove Ball Bearings-Pure Radial Load.. 24

5 Axial Stiffness versus Axial Load - No Radial Load ............... 26

6 Axial Deflection versus Axial Load - No Radial Load ......... .... 27

7 Axial Stiffness versus Axial Load - No Radial Load .............. 2d

8 Axial Deflection versus Axial Load - No Radial Load ......... 29

9 Axial Stiffness versus Axial Load - No Radial Load .............. 30

10 Axial Deflection versus Axial Load - No Radial Load ........... 31

11 Axial Stiffness versus Axial Load - No Radial Load ..... ...... 32

12 Axial Deflection versus Axial Load - No Radial Load .... ..... 33

13 Radial Stiffness for Angular Contact Bearing, Pre Load - SelectedLight 10o - 250 ................ ...... ............ .......... .. 35

14 Radial Stiffness for Angular Contact Bearing1 Pre Load - SelectedLight 0o - 250 ........................ .. ...... 36

15 Radial Stiffness for Angular Contact Bearing, Pre Load - SelectedLight 13 0 2 5 -. ........................................... *... o 37

16 Radial Stiffness for Angular Contact Bearing,Pre Load - SelectedLight 3 0. - 25 ............... ..................... 38

17 Radial Stiffness for Angular Contact Bearing,Pre Load - Moderate

1o = 250 ...................... . . ............. 391i Radial Stiffness for Angular Contact BearingPre Load - Moderate

1o = 250..o ............ ..... ..... ................................ 40

19 Radial Stiffness for Angular Contact BearingPr. Load -Moderate

a 250.................. ... .. 4110

V

FIGURE PC

20 Radial Stiffness for Angular Contact BearingPre Load - Moderateo 25 ............................................. . ..... 42

21 Radial Stiffnea8 for Angular Contact Bearing,Pre-Load - PreferredHeavy oo - 25 ................................... ... ....... 43

22 Radial Stiffnes 8 for Angular Contact Bearing1 Pre Load - PreferredHeavy Oo - 25 .......................... ...................... 44

23 Radial StiffnesS for Angular Contact BearingPre Load - PreferredHeavy di,, S 25i ............... ......... . .................. a. 45

24 Radial Stiffness for Angular Contact BearingsPre Load - PreferredHeavy Sif e 250 .................................................gt

25 Radial Stiffness for Angular Contact Bearing,Pre Load - SelectedLight so - 15-.......................... ..*.. ............. 0 ... 47

26 Radial Stiffness for Angular Contact Bearing1 Pre Load - SelectedLight jo " 150 .................................................. 48

27 Radial Stiffness for Angular Contact BearingsPre Load - SelectedLi 150. 15° 0....................................... ..... 59

28 Radial Stiffness for Angular Contact BearingPre Load - SelectedLght o. 15° .. 0 ...................... 50

29 Radial Stiffness for Angular Contact Bearing. Pre Load - Moderate

30 Radial Stiffness for Angular Contact BearingPre Load - Moderate,so - 15 . ................................................... 0....... 52

31 radial Stiffness for Angular Contact Bearin;, Pre Load - PoderateHev - 15- 15............................... 53

32 Radial Stiffness for Angular Contact Bearing, Pre Load - Proderate

S 150 ...... * .......................... ............ 5633 Radial Stiffnese for Angular Contact Bearing, Pre Load - Preferred

Heavy 0° - 15 ................................... .............. 5534 Radial S t Iffne.8 for Agua Contac t Bearins, Pre Load - Pre ferried

35 Radial Stif fne8S for Angular Contact Bearin•h Pre Load - PreferredHeavy 130 " 15- . ......................... .0.0.......... 0..... 0.... 57

36 Radial StiffnesS for Angular Contact Bearing. Pre Load - PreferredHeavy ° " 15 .............................................. 58

vi

A ..

I

INTRODUCTION

The purpose of this report is to present design data for typical deep-groove andangular contact bearings of the comonly used light and extra-light series. Thedata may be used for ,rartous rotor-ball bearing system designs, in conjunctionwith our (NTI) critical speed and unbalance response programs.

The information is presented in graphical fo-m. It consists of load carrying

capacity, radial and axial stiffness, and load levels.

A complete description is given for all the variables used. A section entitled"Design Requirements" is written to describe various parameters and to presentdesign considerations, guidelines and limitations.

A number of examples on the application of the curves to specific cases are

included.

The analyses used are written in the Appendix, along with a computer program

listing of the calculational procedure.

If any particular case is not covered by the included curves, sore data may be

generated by computer program PWO-182, IlK 1620-60.

I

II

DESIGN REQUIREMENTS

A ball bearing schematic is shown as Fig. A to illustrate standard nmenclature.

Types of Bearings

The single row, deep groove ball bearhg will sustain radial loads and in addi-

tion a substantial thrust load in either direction. Wnen using this type of

bearing, careful alignment between the shaft and housing is essential.

The angular contact ball bearing is designed to support a thrust load in one

direction or a thrust load (preload) combined with a radial load. These bearings

can be mounted singly or, when the side surfaces are flush ground, in multiple,

either face-to-face or back-to-back for all combLnatior,- of thrust and radial

loading. The basic difference between th. two is the larger clearance and

greater shoulder height of the* angular contact bearing. Generally, this will

permit operation with hLgher thrust loads and at higher speeds than the deep

groove bearing.

Load Level

The load levels shown (C/P - 5 and C/P - 10) correspond to normally encountered

Hertz Stress levels of 230,000 and 186,000 psi respectively.

Ball Bearing Damping-

The only available damping information was from non-rotatLng tests on a grease

packed ball bearing (A-2) system. The measured value was in the order of 15-20

pounds sec/in. This should be used only as a "ballpark" since it should be

much higher in the rotating condition, and for larger bearing sizes.

Race Curvatures

SL•ce the question of stiffness and rotor dynamics will be a sajor factor at

high speeds, some design guidelines for this aspect are in order. Normally,

more open curvatures, one piece machined retainers, and gemnrous internal clear

ances are preferred. The normal curvatures are 51.6 percent for inner race and

53 percent for outer. Open curvatures, for high speed range, between 54 percent

and 57 percent for both inner and outer are used. The 57 percent curvature is

widely used and is included in the design discussion.

2

Contact

I Angle00

I d

0 0

IU e

0 sore

ILoad,1RadialLoad,_______P

Fig. A Ball learing Schematic

3

P OWW

7' m

3ll Bearinas - Life

Th. selection of ball bearings for various applications consider such factors as

load, speed, temperature, environment, design, and lubrication. However, the

initial sizing and selection is usually based upon the fatigue rating of the

bearing.

Based upon a statistical distribution proposed by Weibull and the analytical and

experimental work of Lundberg and Palagren the life of the bearing for a given

probability of survival has been found to vary inversely as the cube of thd

applied radial load. For other than radial loading, an equivalent radial load

is defined. A Specific Dynamic Capacity (C) Is defined as that radial load

which will result in a life of one million inner race revolutions with a 90

percent probability of survival. The AFrIA (Anti Friction Bearing Manufacturers

Association) has standardized on the following formula for (C):

0.7 2/3 1.8C = fc(i cos po) 0 7 3 d (1)

where i - the number of rows of balls in any one bearing

- the number of ballsper row

-o the angle of contact

da the ball diameter,"inch

f - a factor dependig on oscillation and materialc

For normal bearing proportions fc - 1500.The life (90 percent probability of survival) at any other radial load or equi-

valent radial load (P) is related to the Specific Dynamic Capacity (C) as

follows:

L - (C/P)" millions of inner race revolutions (2)

It is normally assumed that speed affects life in a linear fashion; that is,

life varies inversely with speed. For a given operating speed of N rpm, the

number of revolutions which correspond to H hours of life is

L - 60 NH revolutions

Bared upon experimental data, equation (3) is too conservative at high speeds.

.4

I- ____________________________________I-

and

3 6H -C) x110I hour (3)

60 N

Catalog ratings are gene.ated in this fashion, usually for some given number of

hours of life with a 90 percent probability of survival. Five hundred hours is

a common catalog rating. Note that a rating at 33-1/3 rpm for 500 hours is the

Specific Dynamic Capacity.

Since this is available in the catalogs of most of the bearing companies, no

further explanation of this aspect is included in this report.

Bearing Centrifugal Loadina

In some instance@, it is desired to estimate the life of a ball bearing at

extremely high speed with little or no externally applied loadin3. In this case*

the fatigue life. is determined by the centrifugal loading of the balls on the

outer ring. (Ref. 4)

The outer ring capaci:y is given by

SA f2° 0.41 1' +7.39 d. 8 -1/3

7) A

where

A - material constant, usually 7140

f - outer race curvature factor (ratio curvature radius to balldiameter)

7 - ball diameter to pitch diameter ratio, d/E

d - ball diameter - inch,

Z - number of balls

R a pitch diameter - inch

The life of the outer ring is given as

3 6Co/Pf. * 901 life in 10 revs.

1~ 5I _ _ _ _ _ _ __ _ _ _ _ __ _ _ _ _ _

where

P .f. , centrifugal ball loading

Pc.f. 5.257 x 107 d3 UI 2 (1-7)2

Effects of Cyclic Loading on Bearing Life

As was previously shown, the fatigue life of a rolling element bearing is defined

in terns of a 90 pFrcant probability of survival. A specific dynamic capacity

C is defined as that radial load which will result in a life of 10 inner race

revolutions with 90 percent survival probability. The 90 percent life at any

other load P is related to the specific dynamic capacity as follows:

L - (C/P)3 106 Rev. (2)

When the load varies in a series of known steps, some equivalent or mean load

is defined as follows:

3 + 3 3 V" N1 + 2 N + n 1

where PV' p2 ' P are loads applied for N1, N2, Wn cycles.

For the case of vibratory loading, an integral form of Equation (6) can be used:

P. " fP3 dN (7)

In the general case, the loading will consist of som steady load Po and a

sinusoidal load Pl sin GA.11

The bearing loading P is given as

S~~P - Po + Pl sin at110 1PmP +(8

Using this in Equation (7) yields the following:

P (P + P smut) 3d(t)(

6

Expansion of Equation (9) gives

1P~ '(P. 3 + 31 2 P O *+3 i2o

3 3 1/3i + •l3 °/'"3c) d at.)] (O

1t+ P" 1 sino• 00 o 4€°8003 o10)

2 2pP P 3~ -t :3 P coo cot + ?

ci 4

.37 0l 1i •os~)) 1 /2

-3 P i 4 P 3 coo Wt + P13co 3 m /

1/3

2 1/3[P1'3 (1 + PC

2 1/3

_3l~( L (12)0 2 o

The results of Equation (12) are plotted in Figure 1 as a function of the cyclic

load ratio.

The life may be found from Equation (2) using the equivalent load

Often the steady state load i is known and the effect of various cyclic loads

is desired. The life due to the steady state load P Is

0 0

* (C ?0 1 13)

V 7

I

-'

1.36

1.34 - -

1.32 - --

1.30--

1.28

1.26 ---- -

1.24 -

1.22 - -

1.20

.'° 1.18

1.16 -

1.14 -

1.12 #

1.10 -

1.08 -

1.06- --

1.02 -

1.00-•0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

? I /

Fig. B Equivalent Load Ratio as aFunction of Cyclic Load Ratio

8.

•.4...

While the life due to the equivalent load Pe is

L - (C/P)3 (14)

LThe ratio of the lives is

0 (C/Po)

The life ratio as a function of the equivalent load ratio is shown in Figure C. i

In summary, the resulLs apply to the following:

1. P1is Po

2. Radial Loading

3. P is unidirectional0

4. P Iis the single amplituda of the cyclic disturbance.

The above can be used with manufacturers' catalog data by roting that these areset up for some given life (usually 500 hours) at various speeds. The corresponding

load is tabulated.

The case where a cyclic load only is applied is sometimes encountered. The

bearing load is then

P P1 sin wt (16)

Equation (7) now becomes

,1/3. (P1 sint) d(wt)(17)

. 3 (-Cos Mt + - cos wt) 1/3

- 0.752 P1

I..

94f

_ _ _ _ _ _ _ _ _q I

1.0 -- ~

0.9-

0.6 8

0.7

0.6

0.5 -

0.4

0.31 - 1

1.0 1.1 1.2 1.3 1.4

p

Fig. C Bearing Lie Ratio AnA function of IquivalcatLoad Ratio

* -- ~~10 ~ '.:~

The ratio of equivalent loading to the single iaplitude of the cyclic loMang

is

, 0.752 16)11

Be~rin8 life is determined from Pa.

As was previously noted, Equation (12) and Figureal were derived'for radial

loading. However, the relations can be adapted for use with thrust loads, if

the thrust load is represented by

T-a +To + aT, (19)

where To is the steady thrust load and Tl is the single amplitude of the cyclic

thrust loadtng. This gives a similar relation to Equation (12) as follows:

S + 20)0T To

Figure B can be used to obtain either a man radial or thrust load. However, In

the case of thrust loading, bearing life must be calculated using an equivalent

radial load with the specific dynamic capacity. For calculation purposes

(preliminary engineering calculations) the equivalent radial load is given by:

""quiv. lad. Load - 0.37 P + 2T (21)

where P is the radial load and T is the thrust load. Either T or P, or both

are replaced by Pa and Tm where cyclic loading is involved. No*r accurate

relationships for the various hearing types are found in the manufacturerse

catalogs or the AFh3V standards.

11

t. 4

II III ,4*

Lubricant Life

In many instances, fatigue life is not the major consideration since the loading

is light. The lubricant is usually the limiting item insofar as life is concerned.

The first consideration is to be sure that lubricant and the lubrication system

are adequate for the speed range.

Unfortunately, there are no exact Suiderules that can be set. However, somae

generalization* are possible with respect to normal applications.

System Speed Limit D xl (bore in me z speed in RiPM)

Grease 250,000 (ribbon retainer)

Oil Level 300,000 (ribbon retainer) 01

Mist 700,000 (machined retainsr)

Jet Oil > 106 (mahined retainer)

Above 300,000 dN, the usual ribbon retainer would be replaced by a machined retainer

of metal or phenolic. For normal temperatures, the phenolic,'etainer is commonly

used. With special greases, retainer design, and light loading, grease lubri-

cation has been used to speeds of 750,000 dli.

With respect to grease lubrication, there is evidence that life is reduced in som

logarithmic fashion with increasing dN value. This is. similar to the effect of

temperature. Figure D shows a typical behavior of life with respect to tempera-

ture. A reasonable rule of thumb is that life is cut in half for each 10C

rise over 1006C.

12

- -I

100

00

103

------------ z:~ I-i-

- ~ I _ _I_ _

III

Description and Discussion of Charts

There are basically three separate sets of design charts included in this report,

name ly:

a. Pure ladial Loaded Bearings (Deep Groove) Contact Angle 0o M 0"

b,, Pure Thrust Loaded Bearings (Deep Groove) Contact Angle Jo W 100

C. Angular Contact Bearings with Axial Preload and Applied Radial

Load 50 "25°, 150

Table I describes the dimensions and symbols used for the deep-grooved ball

bearings. Table IlI contains inforuation pertaining to the angular contact

bearings.

The first set of four charts contains graphs of radial stiffness versus radial

load. Load levels are indicated on the curves. The effects of bearing size and

race curvatures are illustrated.by these four charts. In geatral, a bearing with

tighter raceway curvatures is a stiffar bearing. For example, a bearing with

curvatures of f n .51b, fo .530 is stiffer than the sam bearing operating

with curvatures of f, a f 0 .570, for the sme radial load. Radial stiffness

is higner for a bearing with a larger bore diameter and/or a greater number of

balls. Note, for pure radial load, the linear relationship between log 91

and log P.

The second set of eight charts contains graphs of axial stiffness and axial

deflection versus axial thrust applied load. Load levels are tabulated ir

Table III for these particular bearings undergoing a pure thrust load since

cross plots of C/P will add confusion when reading the curves. Deflection

curves are included to aid in analyzing a double acting thrust bearing set.

The deflection curve for a double acting thrust bearing set is constructed

from the deflection curve of a single bearing by adding increments of deflection

to one bearing and subtracting from the other. The corresponding load

differences equal the externally applied load. A similar observation,as given

14

above for radially loaded bseringa, can be made for the thrust loadel bhawing,

a bearing operating with curvatures of f * .516, fo - .530 is stiffer than thaf

same bearing operating with curvatures of fi a f1. .570, for the sm axial

load. For all practical purposes, however, an average curve may be drawn for

axial Pciffness versus axial Toat for all bearing sizes. In particular, the

bearing with the similer bore and loes balls is less stiff at light loads and

more stiff at heavy load* as compared to the larger bore bearing. There is an

approximate linear rAlationship between log 3A and log T.A

The third set ..of (24) charts contain graphs of radial stiffness versus radial

load. Loa4 levels are indicated on the curves. The effects of bearing size,

race curOatures, initial contact angle, and- axial preload are Illururated by

these" 24 charts. For the sam radial load and axial preload, a bearing operatt

with curvatures of f, a .516, fo a .530 is stiffer than the same bearing operat

with'curvatures of f, a fo - .570. The radial stiffness level is higher for a,

00bearing with a larger bore diameter •and/or a greater number of thalis, and thc

smaller initial contact angle. (1o - 150). In general, the radial stiffness-0

radial load curve for an angular contact bearing is composed of three different

behaving regions. One region' shows the stiffness to be constant with varying

radial load. (This is the light radial load region.) The middle, or moderate

radial load region shows a minimum value for radial stiffness. The heavily

radial loaded region shows a linear relationship between log SL and log P. Th'

third region is similar in behavior to that of the characteristics of a pure r4

loaded deep grooved bearing. The basic cause for this curve hav'-,g three sepal

regions is due to the axial preload. In region one, the axial preloa4 has a

great effect in holding the radial stiffness constant. In region two, where td

applied radial load becomes equal in magnitude to the axial preload, the radial

stiffness tends to decrease with increasing applied radial load to a minimum

value. In the third region, the axial preload has little or no effect, and bh

angular contact bearing reflects the behavior of a pure radially loaded bearii

i.e., a linear log S1 versus log P relationship.

Thus another point one is led to observe is the role of axial preload magnitud

on the three regions of a tyical stiffness versus load curve. Three differaei

preloads are represented in these charts and are tabulated in Table I1. These

I- 15

PJ

preloads are given the umes selected lisht, moderate, ad p'ef~ered heavy. The

effect of increased preload is to increase the region am teod ra sad decrease

region three load range. Thus, the ultimate is a eotant radial stiffness with

varying radial load obtained with an infinite preload. the increased preload

also has the effect of increasing the level of stiffness in regions one and two.

However, it should be noted particularly that the level of stiffness in region

three, for the sam radial load, is the sme for all preload values. This, as

mentioned above, is because the axial preload effect is relieved entirely

above a certain (radial loadOxial preload) ratio. (Approxlmately ViT - 3 for

0 0ISO a 25o and 1/T - 4 for 10o M 15°.

In general, the light and extra light deep grooved bell bearings examined here

will have a radial stiffness ranging from 105 to 2 x 106 for redial loads of

fru 10 to 2,000 lbs. The angular contact bearings will have radial stiffness5 6values from 2 x 10 to 2 x 10 for radial loads of froo 10 to 2,000 lbs. The

deep grooved ball bearings will have an axial stiffness per bearing of from4 6 42 x 10 to 4 x 10 for thrust loads of from 10 to 10 lbe. As in the case of the

preloaded radial bearing, preloading will increase these values of axial stiffness.

16

~~~~~* . . . • • " • • - . .' ? •

0J

%4I in6mn I A % in b Mwl w inj in In Nb% I W% W o

.o ~ o o~~ . . . . . . . . . . o .@

. .% . .% . .% . . . . .4 . .4 5 . % - P -

~~~r u0 en I n A u n in an 40 v4i i n n

'A0 4 c n0 o0 w4u M A i 0 % O~~ 0

o -u

90 a

*1*4

J.J U"' IA IA IA

'0 5 C4 O N Nl % 4 c foD wl %n -. ' .- 4 0 00' D cc40 in in- ~ E ~ 4 - N C4 n v 44 i

.i .,• . . . . . . . .• . .

OR

i n % On a n i i n i n 4C4 (4 0A ccI 4 N N ~ P

14N VI wlO4~ I' 'VI66 inONN L% 9~~~~c WN P.4 6- .4 n im 4 .6 , in i n

0 ~ ~ ~ i ini n ~I

~n-4at-0

00

17

go go

rdP . . . *

0 In In 4J 1 In ;4 :4 :4 InC4 C4 C4C

v4 C4 N

"a4 ID C4 %D# P. 'in In C4 P% Ai m

e) "4 ,- 14 Ci4 ~ 4 # i in

""40.0 4u in at at vi v

"4~C* C4Mi Mi

P- o- #4C 4C4 4C4 0 -4 Of 64 C- V qC'0 an W P..loani-No 004 'It 'Pa 0U MiU 0w ai lk

" " C4 in #4" # 4

in t

~~~ -4 -----------------El UC 606 3~A

*i

Table III Azial Loaded Dep-GQrood kerings

Table of approximate load values corresponding to CLP - S and C/IP? 10 load level4

Bearing Symbol Load (Lb)

S-c/ - 5 C/P " 1o

Al 290 100

B1 600 250

Cl 1100 450

DI 2250 950

11 3650 1500

A2 70 30

B2 175 75

C2 300 100

D2 550 250

R2 950 350

WAl 380 155

111 800 300

CCl 1550 650

DD1 3000 1250

331 5050 2100

AA2 95 50

BB2 200 100

CC2 400 200

DD2 1500 350

312 2550 700

19

I _ _ _ __ _ _ _ _ _

)

PURE RADIAL LOAD

20

It l i " i i ii I I

- - - . - - - - - - a - qI�

- - - - - - - - -

- - -- - - - - -

- -- - - -

- - - - - --- a -

-- \7I *------- p

-- S -- - - - - a

\J. V- - - - - -- a

El:

j------- �. -- I

.0

____ A- - - S - - - - j -* I.

Sw. 0

U

- - - - - - - -, - A- - -__-- - - -__S

______ U a 12A.4

U -

o

U mm-- -- -- .4- p4-- N urn

U

- - - - - - b

.� a - - - - -* '

o 0- p4

21

I- - - - - �

I ----- I

- - - - - - - - I

- - - I __I

Si

- - - - - - - - -- 2- - - - I - - ____

_ -J --- U_ ii

Cl Cd C'4C4 U

- ___ ___ U.w.

IC4

___ b..54

________________ USi

____ - - .3

______ - - .3

o a

- **(i�S�e. S 9 U J(u1/q�) uu.uj�g gwpw�

22

S

J.'.

* -,*�. �* ., . .

I. .� - �*.

PA 0

"a

"d~ 66

- - - - - - -

-I - -23

'44

VAI __V

A1,4- - -N

(ul/ql_ _ soIPzgIwpv

24__ 0"

I

PUll THRUST LOAD

25

.4 �.

I

101_ ... .... . .. --- •- -

o - -.. ..- - . - - I t_ .

,44

-10 .!5 B I -ICI Dg D1

All

I 4

-

I"o

No Radial Load

'26

*2

,% I

BI. ____ 0

-0 -3 -A

D!!

10-4

Load (lb)

Fig. 6 Axial Deflection versus Axial Load

No Radial Load

27

1*

_ _HI

106

'.0 12 --AA

lob

10 5

R28

D2 ~II

1 ~C2•B2--

4 2

104 10 1 a 4 a • '1 2 a 4 a 0 T s o • 1,0 3t 4,

Load (lb)

Fig. 7 Axial Stiffnessewrsus Axial Load

No Rtadial Load

28

-0 -2

00

10 C

Load (lb)

Fig. 8 Axial Deflection versus Axial LoadNo Radial Load

29

106

105

V AA

10 10

10 102 10

Load (lb)

Fig. 9 Axial Stiffness versus Axial LoadNo Radial Load

30

,j*#v,

"-40

100

1ig 10 hxa Dflcio vrusAxa La

1031

-=2

106 /

-Am

qb

U

10 5--- -9200S

SBB2-CC2T-=

Fig. 11 AilSifea essAilLa

3 210 1 102 10 3

, Load (lb)

SFig. 11 Axial S~tffness versus Axial loadNo Radial Load

32

( -

r -

22

£

I. I I

/ 4

10 ----

100

II

I i

0-3 , •0

10 1 a 4 9 6 7 a j.0 2 a a 4 0 0 7 el0 3 4

Load (lb)

Fig. 12 Axial Deflection versus Axial LoadNo Radial Load

33

I

I

RADIAL LOAD WITH AXIAL IWAD

34

" !1

I I

I ~~ rt i

'S4

-41

46 in

(uT/l) sauj;2S l~pa

35_

04

1

C4. x

00' bs

vi

___ql _____T2 1*pU

36-

-4A

4.w 1 p a

4Ilql ssuU~gllwit - 37

- I"

on

C a

-~~~~ai a

ot

(aT/ql) soaau:TTS loTpoll

38

- -~

____ '9

I.14

1V[�F �Vt� I,

iI.a__________ - � --.----.--------.--- 9

________ ________ S

_ -4 __ _____

-� �-. I --

I -

\II I J-K ___ f - I _________

0 4

7 - L � .0

-- - L.. _____--I.-.- -. -- - --------- i-----�-----O.-- * '� I

U

4-----K - -- -- I Uw.-4

-- �.4- � �, * 'U I

- I*aO I�,.IEaa

I - 44'4

I 4I .- I �- 04 S

I -I' C.)

I

-------1�- - -.- I- ("4 p

- 0

4 I-- - - Ii -

_ _ - IK I .2i�-47�.. K-"---- 44 K' 4

U V '4

I_ N

0 0 "a

-7--- 40

ok

-u / l -e u j I -.~p: A

40 -&.p

- - , - h * - - - - a * -

_ _____ - - - - - - - _____ -

------ 3- . -�- -.

.4 ___ - - - - -* - - -4.

As

-- *-iljj--t I

I.,

-. 4

U*6.4

U**I

.4 .4 JU

______________________ .4 4

-- -- S 3Jq3

N0

- .--- .4

- - T - - --- 5.

S I- - - .� - - - -

S U 1.0 9'0 us�o o.4 .4

41

- ..- - -

r4

_ I _"_

UU

______ ___V'

10._ _ _ _

0~ I~4

CA C4*o

424

-X-2

'44

0 C1

- aJ

434

__ __ 0

40

*P4

(uT/l) 9au;J~s ITSF

_ 44

V4 I

CC

A: .4

455

N il I

C..

46~

10 9

00-

U4 x

(u/l asuaI -4 OAg

47.

- - - -- I-.-I----1- 1 0

_________ S

� -� -b 'S

_____-U

aSi be

S

___- I:-- ______ -4 U

__ 0-- ___ ___ --- _____ aDa

____ �DUSi S-

A �aJo . - �.a- ,.1

U

-- - 0Ube 9 6 beU � .3 �

- mu* * me*.4 *�4.� �.* �MI*" 'a'

.,4 I

aD 0

*3

N0 '0

-- ___ ---- - *Si-1 - N- - - - - - - - - U

- - - - - - - - i----- b be

- S

0

-4

L 9

U

* .iLii.L-i ....j_____- - - U

o 0-4(uJ/q�) u.;u;;pS 1� j;prU

48

___ 9)

x V

~~OF

- -. 4 -

0

N ___ __ ___ -

bi N - - -. - - -

-h ___

K------__ __

- Kzy 77�

F U-4

-

UIi

___ -- - -- -I:U'IU'a

.4 0

-4------- --- -�--1.. U

*- -- ___ -4 Ii

6-4 -4

-�-4 -

-�--.-- p1 6

__ ___ ____ -4�0 p1

�4 U

II __ ____ - - ___ .1J�h9 .2 'aU 65

1 @4

I o as

I U '4�

I �

aU .4I-.. - --t 'a C

-4,.

.us___ _____

__

___ ____ - -� __ 0 �

-____

-

U

--4

___ Lfl: -

_____ a

0� - 0

_____

- S he

-- ___ _____ __ __�i:

S a

*0 0

-40

(u�/q�) isau;;ps TUIPrE -,

50

- ---. 4

� m- - � -� -.

41-

cc ...4

14 a

0 c-4,

cc 44

00

-- 4

V 4ad-4(

ii C~4 ON

--- 4--- ssau___ -- - 0

-4 I--_ _ -- -- __ _ - ~-51

C4,

I! --

14a ii

-,4

0 (0

N CC4 -4

V v4

_ _ _ _ a-4.

'IoC

___/q ___m I - 0 cV- ____52

044

~u -0--

00)044

14

-44I.,4

*CG

- - -\ -- ~.--~ '-4'.4

-u7ql -------4-----l- -*pl

- - - 53

I -40

A..

C14 10

I 4'J

4 0 C4

II IN

544

-i Aj 04" 0c-44

040

\ 0r. Q

_ _ -o w_ -4 ii

U,'Id

'IV

C- - ___ '.

Iki _ _-. 4 p 0_________~~~~ V___ I, '________

ii- (uT/qr) hbauul:s vwd

C6

gtoo

1an I 4

40

-A " I - - 0 w

0 Ij.3 ~ 44Cl

U C4

y -4 ~ toI - ~ 4JJA. 0

I -% - >

(ulql esaaa~l lu -

56 ~ d~

44

___ _IV__ 04_. 4z z

____ ____ - - - - - - - -

LiiIn

(uBql I-----T2 - UP

- - *1----57

00

IIOU

0*

4 - - Ic-4- --

qNc w"

22 k

a.cI I 1I5-

-~ ~~~~~ to .- - - - . -

_ - I

I 6-4seui I i - .0o

-- ~ -58

IV

SAMPLE PROBLEMS TO ILLUSTRATE USE OF DESIGN CHARTS

Four particular examples are included in this section:

1. Pure Radial Loaded Bearing

2. Unidirectional Thrust Loaded Bearing

3. Double-Acting Thrust Loaded Bearing

4. Radial Loaded, Axial Preloaded Angular Contact Bearing

1. Pure Radial Loaded Bearing

What are the radial stiffness values corresponding to radial loads of 100 and

700 pounds for a deep-grooved ball bearing with a BORE - .5906 inch and

f, M f - .570?

From Table 1, this bearing corresponds to A2.

From Figure 2:

Radial Load (lb) Radial Stiffness (lb/in)

100 2.88 x 105

700 5.55 x 105

2. Unidirectional ThrusL Loaded Bearing

What are the axial stiffness values corresponding to axial loads of 100 and 700

pounds for the same deep-grooved bali bearing as used in sample problem 1, above?

From Figure 7:

Thrust Load (lb) Thrust Stiffness (lb/in)

100 9.10 x. 10 4

700 3.22 x 105

3. Double-Acting Thrust Loaded Bearing

What is the axial stiffness fnr a double-acting, deep-grooved ball bearing,

thrust bearing set, preloaded to 200 pou.,dsi The bearings are type A-2.

From Figure 8: Read off loads corresponding to equal deflections around

preload of 200 pounds.

59

Inus,

Load (lb) Deflection (in)160 2.8 x 10-3

"200 3.1 x 10-3

240 :.4 x 10-3

The axial stiffness is S LL 240 - 160.3 x 10-3

SA = 267,000 Lb/in

This problem may also be solved using Figure 7 in conjunction with Figure 8.

Thus,

Load (ib) Deflection (in) Stiffness (lb/in)

160 2.8 x 10.3 1.22 x 105

200 3.1 x 19-3 1.4 x 105

240 3.4 x 10-3 1.58 x 105

SA M ES - (1.58 + 1.22) x 105 = 2.8 x 105 lb/in

or S = 2 x 1.4 x 105 . 2.8 x 105 lb/inA

For light loads, i.e., loads less than the axial preload, the load-deflection

characteristics are essentially linear.

4. Radial Loaded, Axial Preloaded Angular Contact Bearing

What are the radial stiffness values corresponding to radial loads of 100 and

700 pounds for an angular contact bearing with a BORE - .5906 inch and

fi M fo M .570. The contacL angle is 15 degrees, and it has a medium axial

preload.

From Table 2, this bearing corresponds to PA2.

60

From Figure 30,

Radial Load (Ib) Radial Stiffness (lb/in)

100 4.45 x 105

700 5.90 x 105

Note: One must be careful in using the charts, in particular for the angularcontact bearing, that the design contact angle and axial preloadingdesign value correspond to the values given in the Fijure legend.

61

4

V

REFERENCES

1. New Departure Engineering Data, Analysis of Stresses and Deflections, Vol. 1,

2, 1946, New Departure, Division of General Motors Corporation, Bristol,

Connecticut.

2. SKF General Catalog, No. 425, 1958, SKF Industries, Inc., Philadelphia,

Pennsylvania.

3. New Departure Handbook - Ball Bearing Catalog, Twenty-third Edition, April

1955, New Departure, Division of G4C, Bristol, Connecticut.

4. Jones, A. B., "hMe Life of High-Speed Ball Bearings", Transactions of the

ASME, July 1952, pp. 695-703.

6

62

VI

APPENDIX

A. Analysis

The theory used for predicting the load carrying capacity, deflections, and

stresses of deep-grooved. and angular contact bearings is that of Refererece l.*

The equations relating load and deflection were differentiated to obtain the

equation for stiffnesas.

A calculational procedure was devised for predicting the maximu ball load,

deflection, stiffness, and inner and outer race stresses, as a function of total

applied load, preload and bearing geometry. This procedure was programmed as

computer program PN0182, IBM 1620-60K.

Three separate cases are treated:

1) Pure Radial Load, deep grooved bearing (Ref. 2,3)

2) Pure Thrust Load, deep grooved bearing

3) Combined Radial Load with Axial Preload, angular contact bearing(Ref. 2,3)

Pure Radial Load

Maximum Ball Load. P0

P - 4.37 * P/n A-10

Radial Deflection, 5N

b - p2/ 3 A-2N 0 o"

where

Co 7.8107 x 10" 6 (0e° + C5)/d 1/3 A-3

and

7 C-o' CB" f(fit fo a, d, ,o) A-4

t *

Sea Referencer.Section

63L ____________________________ ______

Pidial Stiffness, SR

SR - 3/2 P/5N A-5

Compressive Stresses, S0 , St

1507 ) 0 1/3 A-6

Sm 0 fsm(15079)

where f sm f(fip fo, K, d, Ao) A-7

and m- i, or o.

Pure Thrust Load

Maximum Ball Load, P0

P a T A-80 n sinJ3

Axial Deflection, 8H

5 - Bd * A-9H coso

where

B -fi + f 1 A-lO

and01 is found by an iteration scheme, written below.

Applied Thrust (Axial) Load, T

T - nd 2K • sin 81 cos 1 3/2 A-11

where B x 1/2

K - 7"8107(CBo + cBi A-12

and

is found as follows:

64

I

Define the following quantities:

T b Cos o a Cos f5 x2 K 01nd. K

Then Equation (A-11) may be written as

G 2 1/3 a(1x- (--) -b

aL t A-13x

Then y - b(l - x 2 ) 1/3 A-14

It is a known fact that b << 1. Therefore, as a goo4 guass to start the iteration

scheme for solving (A-14), let

y by, + by22 + A-15

where 2

and =222 2 A-16

The procedure is:

1) Calculate yl, Y2 from (A-16) and y from (A-15), knowing a and b.

2) Calculate x from (a-13) knowing y.

3) Calculate y from (A-14).

4) Check y from step (3) with y from step (2).

5) If the values are equal

A M coso (x),

otherwise use an average value for y and repeat steps (2) through (5)

until agreement is obtained.

65

S ___"__ __

Axial Stiffness, I SA 1/ s3 p o 33 2"S B 2 si n1 1 + co 32 A-17

A and BK 2 1ll[coo~ -3os co

Compressive Stresses, So, S

Sm m (15079) 1/3A-6

where

f -m f(fil fo0 j, d. 01) A-7

m i, or o

andP is calculated from (A-8)o

Combined Radial Load and Axial Preload

The same procedure for finding 01 as applied in the pure thrust load case is

applied in this case in order to find 5H' A value of radial deflection 8v,

is assumed then the radial force, EV, is calculated as a function of 8H and

bV

The following definitions are written:

k' !V- h' -- andBd Bd

-1 [ - (sin 0' + h') Ccos0 A-18

* - cos , coo V'> -1

a - • X coo*'< -

Maximum Ball Load, P0

P 0 Kd [ sin 0+ h')* + (c6s 0' + k' co, ) 1) 3/2 A-19

where 0 =0

66

Radial Deflection, 5V; Axial Deflection, 5H

5V k'Bd 8 h' Bd A-20

Axial Preload, T

Calculate from (A-ll) and (A-12).

Radial Load, EV

lV ,nd 2 K fo Ad$

where 2 - -2r '(coo "sin" + h') 2 + (cos •' + K' cos oos2 + k' 1os1) cos

A- [(sin 0. + h') 2 + (coo 10 + k' cos 1)2

and -0•e * A-21

Radial Stiffness, SR

S B ._ fo A cooR ndk K' x cos 0'+ kcos*)

[(sin D + h')2 + (Cos 90 + k' coo *)2J 1/2+I+(coa' + k' co , ) + ho. +h')2. ,• :+ ° . -1o.. j:• sin(.,; h,)2 (cOup+ok oo.q 1 1/+

A-22

Compressive Stresses, Sot Si

1/3

Sm- fam(15079) A-6

67

+ ..... .... .....++7 7.. ...

A

where

fe $ f(fil fo0 E, d, 1) A-7

m - I, or o, and P is calculated from (A-19).

B. Computer Program

A Fortran II computer progrim listing is included in this mmorandum.

The Input Format wr.tten below should be followed when using this program, PHO182

IBM 1620-60K.

Input Format

Card I Identification Card

Anything may be punched in columns 2-72.

Card 2 (6 FlO. 6, 314)

Item

1. BORE, Bore diameter, in.

2. OD, Extreme outer diameter, in.

3. DB, Ball diameter, in.

4. FI, Radius of Curvature of Inner Race

5. F0, Radius of Curvature of Outer .ace

6. BETA, Contact Angle, deg (f - 00 for pure radial load)

7. N, Total number of balls

8. IND, An indicator used to specify either one of three differenttypes of calculations.

IND: 0 Pure Radial Load

IND: 1 Pure Thrust Load

IND: >2 Combined Radial Load - Axial Preload

9. LC, An indicator used to stop calculation procedure

LC: 0 Program returns to Card 1 for more input.

LC: 1 Program stops after computation is completed

68 *

t!

• I

Card 3 (3F10.6, 15)

IND = 0

Item

1. RI, Initial Radial Load, lb.

2. RD, Radial Load Increment, lb

3. RF, Final Radial Load, lb. (Not used in calculation. . - 0.0)

4. M, Total number of radial loads

IND I .

Iterm

1. TI, Initial Thrust Load, lb

2. TD. Thrust Load Increments, lb

3. CONV, A radius of convergence used in the iteration process for cal-culating 01. COIV - .0005 is a typical value.

4. M, Total number of thrust loads

IND> 2

Itern

1. TI, Axial Preload, lb2. TD, Axial Preload Increment, lb (Not used in calculation 7 , 0.0)

3. CONV, a radius of convergence used in the iteration process forcalculating 01.

4. M, Total number of preloads (must be seL equal to one; i.e. M - 1)

Note: The total number of radial loads which will be calculated as afunction of axial preload and radial deflection is equal to thevalue of IND. Thus, IND - 20, the ctlculational procedure willsolve for '20' consecutive radial loads. A maximuA of IND - 24is allowed.

Output Format

The output is self explanatory. All linear dimensions are in inches. All loads

are in pounds. The stiffness units are in lb/in. The stresses are measured in

psi.

69

c BALL EAU fF 1S4t ?S CALCULATICJN ROJTT.14E FOR'C PURE RADIAL9PURE THRUST,(UR CO'lHINECD LOAL)ING INCLLUINro CE,*TRIFUGAL_C PNW,182 3JUNEt64) Stitl AND PO(AoJ)L FOR AJ

0 1 MNS I UN FF(1T74T -C12 - -. -(1-tS(2)DS1

0 12). *

-FF(13)8506

FF14)=.520FF15') .)3

__ FF16)x.54ý;FF(7)x.55ý;

FF111) z.59U____ FF(1 2)x.6)U

-__ B1,) Z.816____bb(2)x.92d

Bo ('5j - 1.oj7Bbi 4)x1 .092

Bbb18) 1.385

OHI 1018l.465bb ( I ) 1 .4981bul 12)x1.525

-- C 21 a i 7d'ý5

CC 14)=1.05

__ __ CC (6 .1.22

CCI 8)=1. 321-__

CCi ,)u1.395CCI 11) * .425CCI 12)xl .45 _

DDO 1)a.85JODD(2)a.968DD(3)=1*085

DDO() p1.260DV(6)xI .341 _

001 7)s3 .413DDl8)al.46L -

W)~ I I (I I aI -%S

D0111) ul*585001 12)a*162 ___

85(1)3.85BS(Z)a.948513 )1.0385(1 4)1.075B5(5)zl1.17BS(6)1&124______8517) .1.30

70

BS(8)sl*35H-1; y I aI &4a

8s(10). 1.44BS( 11Jul.4785(12)81.51CS( 1 -.70CS(2)s.78CS( 31 me86CS(41.o90C515)=u.98CS(6rs.04CSI 7=1.09

CS( 9)o=1. 1

C54 111.1.25

DS(l1)*1.l _____

054 2)slo25

DS(4)u1.l45

DS(51&1.58DS(61.1.65

DS(9)ul.65

OSI 111.1.95DS( 1212.30o

6 READ 100#(REXD 107 BREO)C6T.O~EANIDLPUNCH 111

PUNCH 103PUNCH 1949 dQL9QU).,VrT939SETA~tf'IF (IND-li 1.2.3

1 KtAV 107r.RTvRD9,RF`*M-. PUNCH 106

2 READ 1019 TI*TU*CONV.I4

GO TO 4' IWEAU 1(I071.1o.LAICUP7-11

I(OP=KOP.1IF (KOP-1) 15.15.4

18 AK1(1)n.002AKI(Z)=UQ03

AKI(41a.0o ----.

AK1(S)=.006

AKI (7)=.008

AK1(9)..010

AKI I Ano~l

AKiI 131s.Q1dAKI 114)=oUZVLAK1( 15) *.022

71

161

AK1 ( 18d) L d

AK1 (2ý,As.)35AKI (21) :,j4.AKi (22) .2,

GO TO 4.C P~ktLir.Iw*AHY CALCLJLAMh,,,S

4BETs:90174533#3E:TA

ANxN

(COSbCO!,e (rAET)

CALL TLv fFl ,b9Fr9&h,1*2)CALL TLJ (FI,1.,,Fri .L:.1k)

CALL TLU (F~joviFFsot,91ý')CALL TL'J (rO,'lC9FI-CC,12)CLX.'x-(tv-C )*Z*2.v+6CALL TLI' (FZ)9F,FrFqL-c,,?)

CALL TLU rFQ.CsZoFkoý'912

CALL TLU (rlZu,9FPO-,:'91"CALL TLU (F sLs-o ,l

LiuL).j**J. 433 3333

IF (IND-1) 8.9998 R-kj-RD

R=R+k LP~U4, 3 7*t/AtliP03wP0**.U*333333DNzC*P03*PO3

DRDDN-I1.5*rH/f'6

5 P UN CH I R 9 Po 9 0N OR JL It 51.

--- __20 IP (LC) 6.69159 bz IF -.

____ AVx(b/(CA.J+CDI)/7o81CfL-C6)*3

_AKz SQR TF(IAK.Ar~zAK*DTxT I-T0.

-- AýCL.Sb -

Y11l.Q;A*A/3.0

Y~-.66-667?*A*A*Y1

__ -Do 1,' 18104iTsT+TD

Yxa2ty1+u2*62*y2

72

ETA*ABSF(Y)-AbSFIYY)

--TrTrTWTF TI I1-r* 12

1 1 1UaSTANVC)XS/X)

DH~ml*O/X*SINF(BETI-dETlDk-WB*Ub*DHIZ&DB/E~*X

*-F SG-bl- (Z-CZ) 2*2.0ZFS.I&Zt5-(Zd-ZDI'Ze.U*Z

-DTuA/X-1*O

DTSaSCJRTF (DT)DT6-f__i_5* i.S*XS*XS.0)T*X'X*X/A IIFIIND-1) 16.16911

P03aPU**0.33 3333__ 4FuPFUQ5()'HC0N -

SM(;aFSU*SMI

10 PUNCH 1ý'St T9PO9VH90TDLHSMI9S14

17 Slo(SINb+0H1**42

PUNCH -106'

DO 19 Is1,IND

PHC=(SORTF( 1.0-Sl)-COSU)/9FK

- F(PHCA-1.03 40J.41.4141 PHIs3*1415'fl7

GO TO 3140 PHS*SURTF( 1.C PtIC*PHC3___PH! UATANF(CPHS/PhC 3

Trf014C3 39~3163130'PHI=3.1415927+PHI_

PHID=PHI*57. 29578'5.-DPHISUM~U.0

-- 0 32 J1.*31

CPzCOSF (X)

IF (J -1) 35035933'-- YxO.*5

GO TO 34

34 P=S1+SZ**2

IF (P-1.03 53953954S- 3 Pm41u1.i~E-O6

GO TO 5554 MSRFP1j

55 PM2=PK,**3

73

-.--- l L

SSTnPM2*S2*CP/P -- ~SUM. SUM. SST*YX

S3mCP/ 52

IF(SENSE S#iTCH 21 2352 PUNCM I015953*ST*P9P.4I9SST.5232 SUP4-i5UMS+ bT*b6ST'VX

SUMO(SU?4-Q.!3*SSr .o*PHI./3.14159aeSUMSa ISUMiS-( * 5*SST*ST l*DPHI /3.141 592?IF (SENSE SWITCH 21 5C951

50 PUNCH 1059 SUM9SUMS9PHIbIFK51 PonSQRTFISl+(COS8+FK)O'Z,

-PO=PU**3*AK/AN-

SIMI uP(J3*HCONSMOEFS0*SMISMI*FSI*SMIR ýS-UM A kDRDDN=SUMS*AK/B/DB0NaFK*B*D

19 PUNCH 1'J5,R.POUNoDRD~i)4jS;ItSMlOGO TO 20

15 STOPC FORMAT STATEMENTS

100 FORMAT(72HO

1QZ FORMAT(6F10.69314)1U4 FORMAT (6F10.5ofi)Iw3 FORMAT (7ZrIO oORE (Jol~e bALL UIA* Fil) FIwi COP4T*A

INGTh-N5.o'U~lALLS) I

i~rTTirT.~A.LAL (ALL LCAJ DEFLECTION4 STIFFESS, I*R*STIRESS O.R.STRESS)

1Q7 FORMAT(3FiOd.W9'15)Iv.8 FORMAT472HOTHRUST LOAD BALL LOAD DEFLECTION STIFFNESS I*R*ST

I R E7~. T ARE t 5S)109-FORMAT(62HOTHR.US.T LOAJ T/NODK AXIAL VEIL* ml/lD AXIAL

1ST 1FF.)110 FORMAT(2Fl0e6915)III FORMAT(IHI)

END

74

SUBROUTIN4E TLU IAII5.C.LNjc A NEPENDENT VARTZBEE

C C INOUPENDENT TA8LEc L UL)LIENVET 1Ab' -

C N4 144 OF ENTRtIES IN4 TAOLE

NT 128

NTXIut4

13 IF IM-11-39.39942

GO TO 9942 FT T1TMCd1ej) 45943.44

43 1-1.1GO TO42

44 IF I(dM1-Al ý%,798

7 AwDI /GO TO 99

6 IF (M-NTXZ-1)9*1i.,.1515 RTXI.M

MaM-(M-NTX2) /2

9 NTXI* M6O TOT

8 NTX2=M14 M=INTXI-MIZ*

IF INTX2-NTXIe1I13*1bl.1$

10 DENOu C(.A)-C(IN-1)D IF F. a AHo DIFFDENO*(UtlU4-O:-1u)+QuM

99 RETURN_______END

i7

C. NOenclatur fotr Analyeio

Symbol Definition Units

B Total Curvature, sq. A-10

C° Radial Deflection Constant, Eq. A-3

Deflection Constant of Outer Race, Eq. A-4

ca i Deflection Constant of Inner Race, sq. A-4

d Ball Diameter in.

a Pitch Circle Diameter in.

fi Inner Race Curvature

f Outer Race Curvature0

f Stress Factor for Inner Race, Sq. A-7

f s Stress Factor for Outer Race, Eq. A-7

h' Relative Displacement of Races in Axial Directionsq. A-18 in.

K Axial Deflection Constant, Eq. A-12

k' Relative Displacement of Races in Radial DirectionEq. A-18 in.

n Number of Balls

P Maximum Ball Load lb.0

p Magnitude of Radial Load for Deep Grooved Bearing lb.

SA Stiffness/Bearing in Axial Direction Due to Load lb./in.in Axial Direction

SR Stiffness in Radial Direction lb./in.

S Compressive Stress in Inner Race psi

S Compressive Stress in Outer Race psi

T Axial Load, or Preload lb.

15 Initial Contact Angle des.

Contact Angle after Preload deg.

Operating Contact Angle deg.

Deflection in Axial Direction in.

BK Deflection in Radial Direction in.

6v Deflection in Vertical or Radial Direction in.

EV Magnitude of Radial Load for Angular Contact kg. in.

Angle Measured to a Load Vectoriwithin Loaded Zone deg.of Ball

Half Angular Extent of Loaded Zone of Ball des.(O ,IC' a )

76

*-Jk,~ *. -

Secriy Casifiaton OCUMENT CONTROL DATA -IUD0....wi .ftMomfooi *1 off". how of o~fte"An "dNimn Amme"Wn4 OWN I6 E onea ohm i go o nwA

I ,RMIATIN 0 ACVIVI-V (Ce.pstle. autf.) As. REORT. sOKtjelyy a LAPIIgATION

S. AUTM@(3) (Lose1 maa lNat amo. Andfel

Levi., fid and, Mol w.1kig 8tsu1.y Be

S. REPORT DATE 70. YTOTAL 7m. *V PAgS9 M it. ow mapsfty 1965 15

Ga. COtNTIACT Oil 0MAN? MO.A? 33(615)-1895 9& @Uu01HAT0ft*0 RPOSIT I6UIAO9W)

.T~k No* 304402 S&. T" PONT KOM (A o mbe n~umbes.. o bep Osea.01

tO.AVALAILILIMTAIO0 WOICV QL~fedrequesters iny obtain ooples of~ the epowrftm WC o pammoimoamnt addissemination of this report bW DC Is wetauthori sod, Rlaeto foreig nain al Is not authorised roleas..toN

we frFederal Scientiflo and Technical Informot.iomnfomswry0IAIt UP OT9S Ua. SPONSommo MULTANT ACTvIVTy

IS. ANSYNACT

The contractor shall type In the sae Wahtrect ag

-bat used In the front of the reports

DD I~ JNG1473 _ _ _ _ _ _ _

Secuafty Clasafficadgm

Security Classificationt14 Rev won L AKS LUlmC

'AabriocationDan bearings

Rolling zli -- tRotor-bearing djrninic

INSTRUCTIONSL. ORIGINATING ACTrVIY: Eate the nom* and adrs Iposed by aemmly elsassil~tle maing lam~ mouMeieof the contrctor. subcontractor, prot*@. Depastuest of Do. much auteas" activity or ether orgamifatiom (co"Wnab esther) lesming (1) 1161fe reqaetere My shea coepies of twothe tpt report. w bmr CDC-"2&. REPORT $ECUN9TY CLAUIRFCATION: Eate the evow (2) "FeratW amseenmemest sod dlaasnilastled .6 t0aall security classification of the report indicate whether sab O so sbrs"Rtestricted Dot*" to incided Nothing is to be inm ccer&rps yDCi estete.

AACS ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 WtOn 0 to89ft eI606(3 "U. &Geveremieat g@acie 6646ai PLee ofLace ~ ~ ~ ~ ~ ~ ~ ti witht appkprat twowh r=ultlam direed AenDD.c2b. GROUtJP Autoerntle dowagredi.6 Is specified in DeC DOf- soo s requeSo thmagh

rective 5S 2010 mad Armed Forces Induastrial MeataL Laterthe group numbr. Also. when applicable, @bow that optionalmffarn, have boom used for Group 3 and Group 4 as &other- (4) 11U. L =iltary eagenie awp ebtaim *~Ie of

Plreport directly born DCC. Other qualifted sws&. EWORT TrTLE: Eawe the complete rpett title to aMl shoU request theesgbcapital letter,. Title. to all cases should be umcloseisifedU a amramingul thite ceinot he selected without cleasiftce-tiov. show tite clessifte~atlon im a&H capitols is pereathoee (5) "All distribetlas of this "aer is comehealleiL Qw*Iermedistely foleowing the wjle aled CDC users Sheal reque through4. DESCRIPT IVEt MOTES U appropriate, eate ths tooe of______________reprt e.g. interim, propwse, mumerY, GmRAN2, or (Lol U the repeet ben boom tweasbod to the Office of TssehiaGive the inclusive dates wam' a specific reporting period is Servicee. Depertes of Cnerce. f" ol ae to the piadic, ias&covered. Cat* twos fact sand ster the price. if bnewma,&. AIITVR(OX Eater ths sam~s) of outhot(s) me shows *a K SUmPLMBUARiYa uOTu Use feeassitiamami *sptmesor is the report. Rate lost name. firat same. swage iniiaL toynoeIt military, show reni and breach of service. The KnmowfOthe principal oAutor is as absolute miaisnrn roquiremrneoI 12. OX1OWfU VULTARY ACflV1T: ESote the sagin of

& RPOR DAL fte tbLdAQ f te p" o dl, the deperemeesal preJect of&*c of lobor" r a" peesorifste ("r6 EWRT DTL aterthedateof he rpor am ay. im# I"r) the research and developmmen. Iscinde address.

mofthkya. aot orwath, peoem If amor them one dame appeaseon the repot Use d40e of prablicatiom. 13. AUSTRACT: 3mW, as abotaset giving a brief and taelmm

$0MSnp of theA daCUrest indicatiVe Of the repotw Othe.MVI&TTAL MNOZ*R OF PAGZth The tMal pop eCsat It may also appear oelowber is the bedy of the teehalesl to.should follow we"ra pogialasto procedwee. ioe. enter the Pw- If odditiOSSI -0o is requited, a costioselate sheet shogwaimber of pagve containing Limformatios. 0. 1b a76. MUNBZR OF R&P hMIOCES Ente the toald wor ad ft is blily downrbl. that the abso ot of ciaeeifled rapsw"references cited is the repas.h elaild Ec asahfteabtataaiadwtS. CONTRACT OR GRANT XUE§ZR It aprprsiate. SOW 1- leisto . hemltayseeta l eifaties .6 the Wn

the applicable moobor of the comirect or SPU eat nudewic fenmetien is the pove~sh. repreemeed me (1S). (S3. (C). er (9).the reort wee "rittoTh Iner isom limitatiem on e .6the oshemect Nor-eb. a. &t 3d PROJECT NUM rawe the appeprialp ever ane sme"Gmaed 1600 is beNo Vi t 22S weas.military deperteret idesutifcatios. much "s projectseie. 4.ETV naKy rdaewclal m elhui

~piojct inber. ~ater inhe5, tak ~ Ior shoat preeae that chMreetr1 a repert red map he seed On9&. ORIOIATOWIS RWORTa NUin3RWO) Sawe the OMfi laden 62121 far caftmegims the epwr. Key verds Meet be*WisA repo thnmer by which the docameat will be Idestilied selected so that so security classfllcatnies Amrequired Idsee-sod coatolled by the mriginatlag activity. This w~obr meat Mars. euch me Iqu-irnea model desiqsmaies. bede ane Ioa",irbe "aque to this repot Poleoct code same. pepaphia location. eep be useed as hey96. OTHER RXPORT N1JiER~g): U the reoa ka be weds hat will be filiowed by an laicadmem of hachodal oamp.eassigned soy other roeort aumber (either by the ~ww, OWnt The 5essloeft of lim.s rules, sad weilhs is "90"L.

wrby the opemeor). also nowr this momhes(o)610. AVAELAELITYY/LINKTATION 1001T1CL& Eter sey the

mum lme anfwthsr dlsoemlsatiasi of the rePast, other themft

Secaft ly C wgi~c

i~T