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

Cage Speed and Centrifugal Loading Centrifugal Loading is derived from: Cage Speed is derived from:

Contact Stress for Line Contact

Contact Stress for Point Contact

Contact Stress for Point Contact 90-af D1/2 df/2 rII2 h af L2

FORTRAN CODE COMPLETE 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 Program 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 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

Hertzian Contact Stress Profile for Rib- Roller End Contact Results An example bearing was analyzed with the FORTRAN program and the Hertzian contact stress for the cup, cone and rib were determined: s (psi) _____ Cone Contact Stress _____ Cup Contact Stress x (in) y (in) s (psi) y (in) Hertzian Contact Stress Profile for Cup and Cone Hertzian Contact Stress Profile for Rib- Roller End Contact

Results Optimum cup angles were determined based on bearing speeds (DN). DN (million) Cup Angle (deg) 0.25 40 0.50 0.75 30 1.00 25 1.50 20 2.00 15 3.00 10