High Speed Tapered Roller Bearing Optimization

Brady Walker

3/11/08

Background

The intent of this project is to determine the optimal cup, cone and rib angle for various speeds.

Historically, tapered roller bearings have not been used for high speed applications however they have shown promise if designed properly.

Approach

• Determine Force Balance/Internal Loading• Determine Cage Speed & Centrifugal Loading• Determine Contact Stress for line contact• Determine Contact Stress for point contact• Write program to analyze tapered bearing with

varying geometry• Results/Conclusion

Force Balance/Internal Loading

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:load applied the toequal cup on the forces theof sum theBy taking

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ustAppliedThrQ

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or

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or

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Cage Speed and Centrifugal Loading

2.3 8

1

: tosimplifes then 2.2Equation

2.2 4

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2

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,2.0 into 2.1equation ngsubstitutiby therefore

2.1 4

1

,

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1

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2

2

cmtc

cmtc

t

cmc

dlDF

dlDF

lDm

where

dmF

i

c

md

D

1cos1

2

1

Centrifugal Loading is derived from:

Cage Speed is derived from:

Contact Stress for Line Contact

lb

Q

2

max

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cos

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5.5 racewaysor roller on crown no assumes 1

2

5.4 racewaysor roller on crown no assumes 1

2

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d

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where

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Qxb

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Qb

and

b

y

lb

Q

Contact Stress for Point Contact

ab

Q

2

3max

rf

II

II

II

IIIIII

II

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II

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I

dr

r

rr

rrrr

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Qaa

EE

Qbb

and

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cos2

surface)flat (i.e.

Radius Spherical endRoller

4.6 1111

4.5 0236.0

4.4 0236.0

, toreduce equations esecontact thin bodies steelfor

4.3 11

2

3

4.2 11

2

3

2

1

21

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1

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2

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Contact Stress for Point Contact

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FORTRAN CODE COMPLETEProgram Input

ENTER CUP ANGLE (DEG)

15

d1= 1.50000000000000

r= 18.0000000000000

dm= 9.69999980926514

lt= 1.89999997615814

leff= 1.89999997615814

z= 17

ENTER SHAFT SPEED (RPM)

8000

speed= 8000.00000000000

Thrust= 35000.0000000000

Program Output

theta= 1.57079637050629

v= 3.926990926265717E-002

L2= 19.1035022615163

d3a= 6.96267373831584

phia= 2.408823159920695E-003

h= 4.335798814248844E-002

d3= 7.04793764817241

Dmean= 1.42540634891911

omega= 3426.69725302619

Fc= 1393.84159411333

Qi= 6559.65979968180

Qo= 7954.68305904302

Qf= 878.124719291193

rI1= 18.0000000000000

rI2= 18.0000000000000

rII2= -19.3374410264202

rhoI1 5.555555555555555E-002

rhoI2 5.555555555555555E-002

rhoII1 0.000000000000000E+000

rhoII2 -5.171315059907503E-002

rhosum 5.939796051203608E-002

Rx= 260.253672199079

Ry= 18.0000000000000

astar= 2.77192458942123

bstar= 0.489000261251888

a= 0.307139809064031

b= 5.418309265928676E-002

Qf= 878.124719291193

max sigma= 25194.0320234045

delta= 2.627865758753307E-004

max subsurface shear= 6241.82061708467

depth to max shear= 2.641659139040329E-002

Qimax 172318.497159640

Qomax= 164245.549705140

ResultsAn example bearing was analyzed with the FORTRAN program and the Hertzian contact stress for the cup, cone and rib were determined:

_____ Cone Contact Stress_____ Cup Contact Stress

psi

y (in)

Hertzian Contact Stress Profile for Cup and Cone

(psi)

y (in) x (in)

Hertzian Contact Stress Profile for Rib- Roller End Contact

Tapered Bearing Stress Distribution (2.00 million DN)

0

50000

100000

150000

200000

250000

300000

350000

5 10 15 20 25 30 35 40

Cup Angle (deg)

Cu

p,

Co

ne

Str

ess

(psi

)

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

Rib

Str

ess

(psi

)

Cup Stress

Cone Stress

Rib Stress

Results

DN (million)

Cup Angle (deg)

0.25 40

0.50 40

0.75 30

1.00 25

1.50 20

2.00 15

3.00 10

Optimum cup angles were determined based on bearing speeds (DN).