high speed tapered roller bearing optimization
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
1.0 sin
:load applied the toequal cup on the forces theof sum theBy taking
oo z
ustAppliedThrQ
1.1
sin
sinsin
0sinsinsin
:zero toequaldirection x in the forcesroller theof sum theTaking
i
ffooi
ffooii
QQQ
or
QQQ
1.2
tan
sincos
tan
sincos
0tan
sincos
tan
sincos
:zero toequaldirection y in the forcesroller theof sum theTaking
i
ff
i
oocoo
f
i
oocoo
i
fff
QFQ
Q
or
QFQQ
Cage Speed and Centrifugal Loading
2.3 8
1
: tosimplifes then 2.2Equation
2.2 4
1
2
1
,2.0 into 2.1equation ngsubstitutiby therefore
2.1 4
1
,
2.0 2
1
22
22
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
5.7
cos
5.6 cos
5.5 racewaysor roller on crown no assumes 1
2
5.4 racewaysor roller on crown no assumes 1
2
,
5.3 1035.3
bodies contacting steelfor
5.2 114
,
5.1 12
3
2
1
2
22
2
12
m
omeano
m
imeani
omeano
imeani
II
I
I
d
D
d
D
D
D
where
l
Qxb
EEl
Qb
and
b
y
lb
Q
Contact Stress for Point Contact
ab
Q
2
3max
rf
II
II
II
IIIIII
II
I
I
II
I
I
dr
r
rr
rrrr
Qbb
Qaa
EE
Qbb
and
EE
Qaa
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
2121
3
1
*
3
1
*
3
1
2
22*
3
1
2
22*
Contact Stress for Point Contact
f
D1/2
df/2
rII2
h
f
L2
ff
II
dr
sin22
if
R
I
I
R
i
ilf
r
Da
rh
DL
LD
and
hDd
90
22sin
cossin
2sin2
sin2
cos
sin2
1
max
1
max2
21
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).