EXPERIMENTAL EVALUATION OF STRESSES IN SPHERICALLY … · to provide this stress information by determining experimentally the sur-face stresses in spherically hollow balls through

N A S A C O N T R A C T O R

R E P O R T

N A S A C R - 2 4 4 5

EXPERIMENTAL EVALUATION OF STRESSES

IN SPHERICALLY HOLLOW BALLS

by L. J. Nypan

Prepared by

CALIFORNIA STATE UNIVERSITY

Northridge, Calif. 91324

for Lewis Research Center

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • AUGUST 1974

https://ntrs.nasa.gov/search.jsp?R=19740022849 2020-05-10T07:09:26+00:00Z

1. Report No. 2. Government Accession No.

NASA CR-24454. Title and Subtitle

EXPERIMENTAL EVALUATION OF STRESSES INSPHERICALLY HOLLOW BALLS

7. Author(s)

L. J. Nypan

9. Performing Organization Name and Address

California State University18111 Nordhoff StreetNorthridge, California 91324

12. Sponsoring Agency Name and Address

National Aeronautics and Space AdministrationWashington, D. C. 20546

3. Recipient's Catalog No.

5. Report DateAugust 1971*

6. Performing Organization Code

8. Performing Organization Report No.

None10. Work Unit No.

11. Contract or Grant No.

NGL 05-062-00213. Type of Report and Period Covered

Contractor Report14. Sponsoring Agency Code

15. Supplementary Notes

Final Report. Project Manager, Harold H. Coe, Fluid System Components Division, NASALewis Research Center, Cleveland, Ohio

16. Abstract

An experimental stress analysis was undertaken to evaluate stresses within spherically hollowbearing balls proportioned for 40, 50, and 60% mass reductions. Strain gage rosettes were usedto determine principal strains and stresses in the steel ball models statically loaded in variousorientations. Dimensionless results are reported for the balls under flat plate contact loads.Similitude considerations permit these results to be applied to calculate stresses in hollow bear-ing balls proportioned to these mass reductions.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Hollow balls; Ball bearings; Stress analysis Unclassified - unlimitedCategory 15

19. Security Classif. (of this report) • 20yiSecurity Classif. (of this page)

Unclassified Unclassified21 . No. of Pages 22. Price*

48 $3. 25

* For sale by the National Technical Information Service, Springfield, Virginia 22151

TABLE OF CONTENTS

Page

I. Summary 1

II. Introduction 2

III . " Models 4

IV. Instrumentation 6•

V. Results and Discussion 8

VI. Comparison with Theory 11

VII. Conclusion 13

VIII. References 14

111

LIST OF FIGURES

Figure Page

1. Model Dimensions 15-17

2. Strain Gage Locations 18-20

3. Photograph of Models Used 21

4. Photograph of Model Positioned for Testing 22

5. Circuit Diagram of Instrumentation 23

6. Photograph of Apparatus Used 24

7. Stresses in 40% Mass Reduction Model 25-26



IV

LIST OF TABLES

Table Page

1. Strain Gage Locations 31

2. Strain Data, Principal Strains and DimensionlessPrincipal Stress Coefficients for 40% Mass Re-duction Model. 32-35



5. Dimensionless Principal Stress CoefficientsCalculated from Theoretical Solution. 44

I, SUMMARY

An experimental stress analysis was undertaken to evaluate stresses

within spherically hollow bearing balls proportioned for 40, 50, and 60%

mass reductions. Strain gage rosettes were used to determine principal

strains and stresses in the steel ball models statically loaded in various

orientations.

Dimensionless results are reported for the balls under flat plate

contact loads. Similitude considerations permit these results to be

applied to calculate stresses in hollow bearing balls proportioned to

these mass reductions.

II. INTRODUCTION

Aircraft gas turbine engines currently operate in a speed range of

1.5 to 2 million DN (bearing bore in mm times shaft speed in rpm). It

is estimated that engine designs of the next decade will require bearings

to operate at DN values of 3 to 4 million. In this DN range, the re-

duction in bearing fatigue life due to the high centrifugal forces de-

veloped between the rolling elements and outer race becomes prohibitive.^ •

To solve the problems of reduced fatigue life in high-speed ball

bearings various methods of reducing centrifugal force have been pro-

posed. One of these is to reduce the ball mass by welding forged hollow

hemispheres and finishing the spheres in a manner similar to solid

balls. Full-scale bearing tests with spherically hollow ball bearings

have demonstrated that operation is possible. ' Fracture of spherically

hollow balls has also been experienced during the operation of the full-

scale bearings.

Analysis of the failures experienced, and the effect of changes in

ball characteristics and configuration on high speed operation has been

handicapped by the difficulty of application of theory to predict stresses

existing in the balls under bearing loads and centrifugal forces applied

at various locations on the balls. It is the object of this investigation

to provide this stress information by determining experimentally the sur-

face stresses in spherically hollow balls through the use of oversize ball

models of sufficient size to instrument with strain gage rosettes so that

conventional experimental stress analysis techniques may be employed to

Numbers in parentheses designate references at end of report.

obtain this information.

Strain gage techniques have been used to determine the surface stress

distribution in spherically hollow balls proportioned for mass reductions

of 40, 50 and 60 per cent.

III. MODELS

Actual bearing balls dynamically loaded in a full scale ball bearing

would be difficult to instrument for experimental stress analysis. The

models used in this study were selected for ease of fabrication and in-

strumentation. Hemispheres were turned from mild steel bar stock with

a radius cutting tool to a 63.5 (2.5 in) radius spherical contour, and

turned to a spherical internal radius calculated to provide the desired

mass reduction of 40, 50 and 60 per cent. The hemispheres were-strain

gaged inside and out with internal leads brought out through a 7 mm

(0.27 in) hole in the hemisphere opposite the gages. Figure 1 gives

model dimensions. The mild steel materials simplified metal cutting.

Its lack of hardness was not a problem as care was taken to insure that

strains were always within the elastic range. The 127 mm (5 in) model

size seemed to be compatible with available 1 mm gage length strain.gage

rosettes and proved easy to position and load in a unversal testing

machine. TML ZFRA-1 (1 mm gage length) 45 strain gage rosettes were

mounted on the models in locations shown in Figure 2. The hemispheres

were bonded with epoxy adhesive. Evaluation of strain gage data seemed

to indicate that the bond line transmitted forces and moments sufficiently

well that data from rosettes nearest the bond line was indistinguishable

from the data of rosettes more distant from the bond line.

The rosettes were mounted with one strain gage of each rosette aligned

along a common great circle of one hemisphere over a quadrant. Other

strain gages on the rosette backing were then automatically aligned at

45 and 90° to the great circle. A line of rosettes was thus established

along a great circle on the interior and exterior of the model.

Loads were applied statically at four load points by repositioning

the model each time after the rosette indications had been recorded. As

seen in Figure 2 these load points were midway between the external

rosettes. Loads were carefully limited to protect the models while

still providing a measurable strain gage response. The models were

positioned by eye to provide load over the desired load point marked

on the outside of the model. Figure 3 shows the models used, and Figure

4 shows a model positioned in the testing machine. Strains proved to

be very sensitive to load orientation for the most highly loaded gages.

IV. INSTRUMENTATION

A Baldwin-Lima-Hamilton Model 120 strain indicator was used to power ,

a Wheatstone strain gage bridge incorporating a temperature compensating

strain gage as one of the bridge arms. The strain indicator scope output

jack was used to drive a Mosely 7000 A XY plotter to amplify and record

the strain indicator signal. The recorder pen deflection was found to be

linear with strain indicator unbalance, and the recorder could be calibrated

so that 25.4 mm (1 in) of pen deflection corresponded to 100 micro mm/mm

(100 micro in/in) of strain indicator unbalance. The recorder then could

provide a + 190.5 mm (t 7.5 in) pen deflection and record for a t 750 micro

m/m (t 750 micro in/in) strain unbalance of the bridge by the active strain

gage. Standard commercial Baldwin-Lima Hamilton and Budd switch and balance

units were used to switch individual gages of the rosettes to the strain

indicator and to provide initial zero adjustment for each gage. As each

gage was switched to the strain indicator and the gage unbalance deflected

the pen in the "X" direction a record was made by deflecting the pen 2.54 mm

(.1 in) in the "Y" direction.

These records could later be read to .25 mm (.01 in) so that the re-

solution of the recording system was 1 micro m/m (1 micro in/in). Successive

records made over the gages of a model while the model was undisturbed in the

testing machine at constant load indicated an overall repeatability of t 5

micro mm/mm (t 5 micro in/in) for the overall instrumentation system under

this condition.

When a supposedly identical series of data records were taken on different

days deviations of + 60 micro m/m (t 60 micro in/in) could occasionally

be detected. These were attributed to difficulty in obtaining identical

model-load orientation, and strain gage and adhesive hysteresis effects

superimposed on the above switch contact-strain indicator-recorder

variations. A technique of averaging data was successful in reducing the

effect of this variation.

Figure 5 is a circuit diagram of the instrumentation. Figure 6 is an

overall view of the physical arrangement of the apparatus.

V. RESULTS AND DISCUSSION

Strains read from the recorder charts were used to compute principal

strains and stresses for each rosette. These are given in Tables 2, 3,

and 4.

In these tables epsilon A is the latitudinal strain along a great

circle of the model, read from the output of a strain gage mounted along

the great circle. Epsilon B is the strain 45° to epsilon A, and epsilon C

is the longitudinal strain, read from the strain gage mounted at 90° to

epsilon A. The data reads from the top down as the rosette closest to the

symmetry point of the strain gaged hemisphere downward toward the joint of

the sphere.

Epsilon 1 and epsilon 2 are the computed principal strains. All strains

are given in micro m/m (micro in/in) with epsilon 1 always being the alge-

braically larger (most positive) of the principal strains.

The principal strains were calculated from the measured strains using

equations from Dally and Riley. '

- ~? vc a - e r; ~ r ^ ' - R ~ > ~ ' f l ~ " ' r ' mL H I, D H I/ ' V U

Principal stresses were then computed from:

E /e + £ i°1 = 777 (£1 + v € V (2)

- (£ + £. } (3)2 1 - v2 2 ]

with values for modulus of elasticity, E, of 207 X 109 N/m2(30 X 10 Ib/in )

+ 1

and a Poisson's ratio,v, of 0.3 These were then used to detrmine dimen-

sion! ess stress coefficients from:

V — 'Kl - ~P (4)

K2 = - (5)'

where: R = the outer radius of the model

P = the load required to produce the measured strain.

Loads used were limited to protect the models by observing the out-

put of the gages and stopping the loading when a significant indication

was observed. There was some uncertainty as to the adequacy of the epoxy

joints in the models, however, the data obtained proved to be consistent

when a number of measurements were averaged, and the loads reported in

Tables 2, 3 and 4 were found adequate for the purposes of the study.

Five independent sets of data were obtained at each load location over

a number,of days. The model was removed from the universal test machine

and repositioned for each data determination. The strain data measured

from the chart records was analyzed by computing the mean values of the ;5

determinations of e., EB and e-; the deviation of each determination from

the mean and the standard deviations of e., £„* and e~ . Deviations were

scrutinized carefeully, and chart records reexamined where deviations were

greater than 15 micro m/m. While this helped to eliminate large errors

in observations, there were still standard deviations of up to 28 micro m/m

in measurements of 144 micro m/m magnitude. The largest variation

occured where rosettes were almost directly below the load location,

and are attributable to the great variation in dimensionless stress co-

efficient with angular orientation to the load, as may be seen in Figures 7, 8 &

9. In spite of large differences in independent data determinations for

some rosettes the averaging of 5 data sets and use of all available

data gave smooth curves that were consistent with the theory of elasticity(4)series solution given by Go1eckiv ' and discussed in Section VI of this

report.

While only 5 rosettes inside each model and 3 rosettes on the outside

of the models were used, the model symmetry and the interlacing of the

indicated stress coefficients for the load locations used permitted the

curves of Figures 7, 8 and 9 to be drawn. Stress coefficients given in

Tables 2, 3 and 4 were plotted separately for each load location and then

transferred to a single sheet with the aid of a light table.

From Figure 7, 8 and 9 it may be seen that maximum stress coefficients

are 43.1, 75.9 and 142.8 for the 40, 50 and 60% mass reduction balls.

It would seem then that stresses in a 40% mass reduction ball will be

0.568 of those in a 50% mass reduction ball, while stresses in a 60%

mass reduction ball will be 1.88 times those in a 50% mass reduction ball,

all other things being equal. If centrifugal force is the principal

loading factor on the balls a change from 50% to 40% mass reduction balls

would increase load by 6/5 but reduce bending stresses by a factor of

0.682. Conversely going to a 60% mass reduction ball from a 50% mass

reduction ball would reduce centrifugal force by 4/5, but increase stress

by a factor of 1.5. Contact stresses will still be greatest for the

40% mass reduction case.

10

VI. COMPARISON WITH THEORY(4)Golecki v ' has given a theory of elasticity series solution for the

state of stress in "The Sphere Weakened by a Concentric Inclusion of

Different Elastic Properties Under Concentrated Loads". Pih and Vanderveldt

have shown that for spheres with ratios of internal to external diameter

of .25 and .33, the series is oscillatory for radial distances greater

than 0.5 times outer radius and actually diverges near the outer boundary.

The series was found to converge for the interior surface stresses of

spheres proportioned for 40, 50 and 60% mass reductions. It was necessary

to compute and sum 40 terms of the series before the last term of the

series became sufficiently small in the worst case. Table 5 gives the

theoretical values of stress coefficients computed for the three balls

at 10 degree increments in angle 0, the angle from the applied force

The coefficients given by the elasticity solution were used to plot

the curves of Figures 7, 8 and 9 on which the experimentally determined

non-dimensional stress coefficients are superimposed. Agreement between

the curves and elasticity solution appears to be good.

Rumbarger ^ ' reported the result of a finite element computer analysis

of a hollow ball contacting a flat plate. His ball model had a diameter

of 25.4 mm (1 in), a wall thickness of 2.03 nun (.08 in) and was loaded to

4,450 N (1000 Ib). He reported calculated interior bending stresses

of 304,800 N (150,000 psi) directly under the load and 34,500 N (5,000 psi)

at 90° to the load. The radius ratio for this case was .84.

From this it may be inferred that the ball was proportioned to a

59.3% mass reduction and that dimensionless stress coefficients would be

-117.8 and 3.9 at 0° and 90°. These compare with values of -138.1 and 3.09

1 1

interpolated between Golecki's series solutions for a 59.3% mass reduction

ball. The values are of an appropriate magnitude, but do not agree as

well as do the experimental results reported here. The reason for the

lack of agreement between Rumbarger and Golecki is not clear.

12

VII. CONCLUSION

The stress distribution in spherically hollow balls proportioned

for mass reductions of 40, 50 and 60 per cent has been determined. A

40% mass reduction ball should experience bending stresses due to cen-

trifugal loading that are 68% of those experienced by a 50% mass re-

duction ball.

REFERENCES

1. Anderson, W.J., Fleming, D.L., and Parker, R.J., "The SeriesHybrid Bearing - A New High Speed Bearing Concept", Journal ofLubrication Technology, Trans. ASME, Series F, Vol. 94, No. 2,April, 1972, pp 117 - 124.

2; Coe, H.H., Parker, R.J., and Scibbe, H.W., "Evaluation of Electron-Beam Welded Hollow Balls for High-Speed Ball Bearings", Journalof Lubrication Technology, Trans. ASME, Series F., Vol. 93, No. 1,Jan., 1971, pp. 47 - 59.

3. Dally, J.W., and Riley, W.F., Experimental Stress Analysis, McGraw-Hill Book Company, New York, 1965.

4. Golecki, J., The Sphere Weakened by a Concentric Inclusion of DifferentElastic Properties Under Concentrated Loads, Archiwum MechanikiStosowanej, Vol. 9, 1957, pp 301 - 317.

5. Pih, H., and Vanderveldt, H., Stresses in Spheres with ConcentricSpherical Cavities Under Diametral Compression by Three-DimensionalPhotoelasticity, Experimental Mechanics, May 1966, pp. 244 - 250.

6. Rumbarger, J.H., Herrick, R.C., and Eklund, P.R., Analysis of theElastic Contact of a Hollow Ball with a Flat Plate, Journal ofLubrciation Technology, Trans. A.S.M.E., Series F., Vol. 92,No. 1, Jan., 1970, pp. 138 - 144.

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LOAD LOCATION 1

LOAD LOCATION 2

/\ LOAD LOCATION 3

^7 LOAD LOCATION 4

—(^—THEORETICAL

FIG. 7a DIMENSIONLESS PRINCIPAL STRESS COEFFICIENTSFOR 40% MASS REDUCTION MODEL INTERIORLOAD LOCATIONS SHOWN IN FIGURE 2a.

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TABLE I

ROSETTE ANGULAR LOCATION ON GREAT CIRCLE

FROM CENTER OF HEMISPHERE

40% MASS REDUCTION MODEL

Interior

RosetteNumber

12345

50%

Interior

RosetteNumber

12345

60%

Interior

RosetteNumber

12345

AngleDegrees

022.0*33.443.3

54.2

MASS REDUCTION MODEL

AngleDegrees

0

17.234.451.768.4

MASS REDUCTION MODEL

AngleDegrees

-4.99.224.241.3

54.1

Exterior

Exteri or

RosetteNumber

123

Exterior

RosetteNumber

123

AngleDegrees

0°3060

AngleDegrees

0°

30

60

* Six rosettes were mounted in the 40% mass reduction model, however, a rosetteat 11° was found to be inoperative.

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TABLE 5

DIMENSIONLESS PRINCIPAL STRESS

COEFFICIENTS CALCULATED FROM THEORETICAL SOLUTION

FOR INTERIOR SURFACE STRESSES

K = a : P/TrRo2

Mass

9

0

10

20

30

40

50

60

70

80

90

Reduction

Kl

-43.12

- 9.64

10.70

13.32

11.80

9.30

6.90

5.06

3.94

3.57

40%

K,

-43.12

-24.89

- 9.67

- 4.60

- 3.09

- 2.79

- 2.87

- 3.02

- 3.12

- 3.15

50%

Kl

-75.95

-4.81

19.01

19.84

15.94

11.37

7.61

5.04

3.59

3.13

K,

-75.95

-34.20

-11.52

- 5.47

- 4.01

- 3.84

- 3.91

- 3.94

- 3.92

- 3.89

60%

Kl

-142.82

7.07

33.20

30.04

21.26

13.38

8.03

4.97

3.51

3.09

K2

-142.82

- 46.99

- 14.25

- 7.23

- 6.08

- 5.79

- 5.48

- 5.13

- 4.70

- 4.48

NASA-Langley, 1974 F.-7976

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