The ratio of the density of the lubricant leaving thebearing to the density of the lubricant entering the bearing The unit load supported by a parallel-surface thrust bearing The approxi
Trang 1The difference in temperature (T) of the bearing
and of the cooling medium can be found from the
equation
The difference between the bearing-wall temperature
tb and the ambient temperature ta, for three main
types of lubrication by oil bath, by an oil ring, and
by waste pack or drop feed
BEARING CAP
The bearing cap thickness
Hd¼ðT þ 18Þ
2427k ðLdÞ
Customary Metric ð23-80cÞwhere Hdin kcal/s,ðLdÞ in m2,T in 8C values of
k are as given inside parentheses under
Eq (23-80a) for US customary system unitsand values of k for customary metric unitsalso given under Eqs (23-80a) and (23-80b)ðT þ 18Þ2¼ K0Pv SIðMetricÞ ð23-81Þwhere P in N/m2(kgf/mm2), v in m/s, and T in K(8C)
K0¼ 0.475 (4:75 106) for bearings of light tion located in still air
hc¼
ffiffiffiffiffiffiffiffiffiffi3Wa
0 20 40 60 80 100 120 140 160
3 2 1
1 - Thin shell not attached to large radiating mass
3 - Well ventilated bearing
2 - Average industrial bearing, unventilated
DESIGN OF BEARINGS AND TRIBOLOGY 23.55
Trang 2The deflection of the cap
The thickness of cap from Eq (23-71)
EXTERNAL PRESSURIZED BEARING OR
HYDROSTATIC BEARING: JOURNAL
BEARING (Fig 23-47)
The pressure in the lower pool of quadrant 1
(Fig 23-47)
y ¼ Wa34ELh3 c
ð23-83Þ
hc¼ 0:63a3
ffiffiffiffiffiffiffiffiffiWELy
s
ð23-84Þwhere the deflection should be limited to 0.025 mm(0.001 in)
Still air
Moving air
Oil ring
Still air
Moving air
Oil bath
Still air Moving air
Trang 3The pressure in the upper pool of quadrant 3 (Fig.
23-47)
The pressure in the left pool of quadrant 2 (Fig 23-47)
The pressure in the right pool of quadrant 4 (Fig
23-47)
The flow of lubricant through the lower quadrant 1 of
the bearing from the manifold
P0
Po
ð6:283 þ 3:425"2Þ ð23-87bÞ
where
K4¼18
P0
Po
ð6:283 þ 3:425"2Þ ð23-88bÞ
Constant pressure oil manifold
1
L 3
Trang 4The flow of lubricant through the left quadrant 2 of
the bearing from the manifold
The flow of lubricant through the upper quadrant 3 of
the bearing from the manifold
The flow of lubricant through the right quadrant 4 of
the bearing from the manifold
The total flow of lubricant through quadrant of the
bearing from the manifold assuming P2¼ P4¼ P0
(good approximation)
The flow factor in Eq (23-81b)
The external load on the hydrostatic journal bearing
The load factor
The pressure ratio connecting the dimensions of the
bearing and its external resistances
d4Po48l1
W ¼ ðP1 P3Þ
A þA02
o
A þA02
FPFWð23-95Þwhere FPFW¼ load factor given by Eq (23-95)
Po
P0¼ 1 þ 6
d
Trang 5IDEALIZED SLIDER BEARING (Fig 23-48)
Plane-slider bearing
The pressure at any point x
The load carrying capacity
The resultant shear stress at any point along the slider
(Fig 23-48)
The shear stress at any point on the surface of the
moving member of the bearing (i.e., slider at y ¼ 0)
(Fig 23-48)
The shear stress at any point on the surface of the
stationary member of the bearing (i.e., shoe at
DESIGN OF BEARINGS AND TRIBOLOGY 23.59
Trang 6The frictional force on the moving member of the
bearing (i.e., slider)
The frictional force on the stationary member of the
bearing (i.e., shoe)
The coefficient of friction
The distance of the pressure center from the origin of
the coordinates, i.e., from the lower end of the shoe
(Fig 23-48)
Pivoted-shoe slider bearing (Fig 23-48 and
Fig 23-52)
The load-carrying capacity
The frictional force on the moving member of the
bearing (i.e., slider)
The frictional force on the stationary member of the
bearing (i.e., shoe)
22
266
3775B
Trang 7The coefficient of friction
The distance of the pivoted point from the lower end
of the shoe (Fig 23-39), i.e., the distance of the
pres-sure center from the origin of the coordinates
DESIGN OF VERTICAL, PIVOT, AND
COLLAR BEARING
Pivot bearing (Figs 23-49, 23-50, and 23-53)
FLAT PIVOT
The total axial load on the flat pivot with extreme
dia-meters of the actual contact d1and d2
The friction torque based on uniform intensity of
pressure with extreme diameters of the actual contact
d1and d2
The friction torque based on uniform wear with
extreme diameters of the actual contact d1and d2
The power absorbed by friction with d as the diameter
of flat pivot bearing
CONICAL PIVOT
The friction torque based on uniform intensity of
pressure with extreme diameters of the actual contact
d1and d2
The friction moment which resists the rotation of the
shaft in a conical pivot bearing for uniform wear
The loss of power in vertical bearing
¼FmP
W ¼h2B
16
Take CPfrom Table 23-17 for various values of q
xx ¼
ð1 þ qÞð3 þ qÞ lnð1 þ qÞ qð2:5q þ 3Þqðq þ 2Þ lnð1 þ qÞ 2q2
Bð23-111ÞThe ratiosxx=B are taken from Table 23-17
Wsin
d3 d3
where 2 ¼ cone angle of pivot, deg
Mt¼ Wsin
Trang 8If the journal and the bearing are eccentric and the
distance between their axes is ", the power loss is
calculated from formula
P¼ 2:35 1041d2Ln2
Customary Metric ð23-118bÞwhere Pin hpm,1in cP, d and L in cm, and n inrpm
P¼ 2:35 1071d2Ln2
Customary Metric ð23-118cÞwhere Pin hpm,1in cP, d and L in mm, and n inrpm
P¼ 2:35 1060d2Ln2
Customary Metric ð23-118dÞwhere Pin hpm,0in kgf s/m2
, d and L in m, and n
in rpm
P¼ 2:35 1030
d2Ln2Customary Metric ð23-118eÞwhere Pin hpm,0in kgf s/m2, L and d in mm, and
n in rpm
P¼3:83
1d2Ln2
USCS ð23-118fÞwhere Pin hp,1in cP, d and L in in, and n inrpm
Customary Metric ð23-119bÞwhere Pin hpm,1in cP, d and L in mm, and n inrpm
P¼ 2:3 106 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2Ln2
1 ð2"Þ2q
Customary Metric ð23-119cÞwhere Pin hpm,0in (kgf s/m2
Trang 9Collar bearing (Fig 23-51)
The average intensity of pressure with i collars
The friction moment for each collar for uniform
The friction power in collar bearing
The coefficient of friction for collar bearing
Allowable pressure P may be taken so that Pv value
for v ranging from 0.20 to 1 m/s (50 to 200 ft/min)
Mte¼13
Wi
d1þ d24
where P in Pa and v in m/s
Pv 0:0715 Customary Metric ð23-127bÞwhere P in kgf/mm2and v in m/s
DESIGN OF BEARINGS AND TRIBOLOGY 23.63
Trang 10PLAIN THRUST BEARING (Fig 23-50b)
Recommended maximum load
Approximate power loss in bearing
Lubrication flow rate to limit lubricant temperature
rise to 208C
Thrust bearing
Parallel-surface thrust bearing (Figs 23-51 to 23-52)
The pressure at any point along the bearing
W ¼ K1ðd2 d2Þ SIðUSCSÞ ð23-128Þwhere K1¼ 0:3 ð48Þ, W in N (lbf), d1and d2in mm(in)
P¼ K2
d1þ d22
n0W SIðUSCSUÞ ð23-129Þwhere K2¼ 70 106¼ ð11 106Þ
Pin W (hp), n0in rps, and W in N (lbf)
where K3¼ 0:03 106(0.3), Q in m3/s (q.p.m),and Pin W s (hp)
Refer to Table 23-8 for P and Table 23-6 for Pvvalues
; x1¼x
x y
d w
z
r
x ω
z
U
FIGURE 23-51 Parallel-surface thrust bearing.
Trang 11The ratio of the density of the lubricant leaving the
bearing to the density of the lubricant entering the
bearing
The unit load supported by a parallel-surface thrust
bearing
The approximate formula for unit load supported by
a parallel-surface thrust bearing
The pressure distribution along a tilting-pad bearing
of infinite width (Figs 23-48 and 23-52)
0¼ 2 1
¼ 1 þ a1
where a ¼ constant, a= 1¼ 0:0004, and t1and t2are the temperatures in8C corresponding todensities 1and 2, respectively
Trang 12The unit load supported by a tilting-pad bearing of
infinite width (Fig 23-52)
OIL FILM THICKNESS
The thickness of oil film in a parallel-surface thrust
Comparison of load capacities of tilting-pad and parallel-surface-type of bearings
Temperature rise through bearings, 8C 0 K LP3 K lt (for h 0 ¼3) Relative load capacity, K LP3 =K lt
Trang 13For properties of lubricant bearing materials and
applications, conversion factors for viscosity,
kine-matic and Saybolt viscosity equivalents and conversion
tables for viscosity equivalent
COEFFICIENT OF FRICTION
The coefficient of friction in case of a parallel-surface
thrust bearing
Another formula for coefficient of friction in case of a
parallel-surface thrust bearing
The coefficient of friction for a tilting-pad bearing of
infinite width
HYDROSTATIC BEARING: STEP-BEARING
(Fig 23-53)
The pressure in the pocket supplied from external
source to support the load
The load-carrying capacity
The rate of flow of lubricant through the bearing
Power loss in bearing
¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi16KLP3p
3771=2
m ¼ h1=h2¼ film thickness ratio ð23-141bÞ
Po¼8W lnðd2=d1Þ
W ¼ oðd2 d2Þln
in rps
P¼ 8:3 104 n0
16hðd4 d4ÞCustomary Metric ð23-145bÞwhere Pin hpm, in kgf s/mm, h, d1, and d2in
mm, and n0in rps
DESIGN OF BEARINGS AND TRIBOLOGY 23.67
Trang 21SPHERICAL BEARINGS (Fig 23-54)
Equivalent bearing pressure (Fig 23-54)
Maximum bearing pressure if an average bearing life
of 105number of oscillations is to be expected
Bearing life (Fig 23-54)
HARDENED STEEL
BRONZE
HARDENED STEEL
As a rule ϕ <8
FIGURE 23-54 Spherical bearings
Courtesy: Neale, M J., Tribology Handbook, Newnes and
Butter-worths
p ¼W
2
r þ 6W2 a
3
where
L ¼ bearing life, i.e average number of oscillations
to failure assuming unidirectional loading
f ¼ life-increasing factor depending on periodical lubrication
re- 10–15 for hardened steel on hardened steel
1 for PTFE fiber or impregnated metal on dened steel
har- 5–10 for d > 0:05 m bronze on hardened steel
po¼ maximum allowable bearing pressure, assumingunidirectional dynamic loading and no re-lubri-cation
nl¼ average number of oscillations to failure ¼ 105
nr¼ recommended interval between re-lubrication innumber of oscillations
< 0.3nlfor hardened steel on hardened steel
< 0.3nl(usually) for bronze on hardened steel
a The figures given above are based on dynamic load conditions For static load conditions, where the load-carrying capacity of the bearing is based on bearing-surface permanent deformation, not fatigue, the load capacity of steel bearings may reach 10 p o
and of aluminum bronze 5 p o
Ability to carry alternating loading is 1 :7 p o for metal contact; and is reduced by 0 :25 p o for DTFE fibre on hardened steel.
DESIGN OF BEARINGS AND TRIBOLOGY 23.75
Trang 22Load carrying capacity of spherical step bearing
tanð 2=2Þtanð 1=2Þ
W ¼24Q
3d2
2þ 3e e3ð1 e2Þ2
ð23-148Þ
Trang 2323.2 ROLLING CONTACT BEARINGS1
SYMBOLS
a1; a2; a3 life adjustment factors, Eq (23-185a), (23-185b)
B width of bearing, m (in)
c permissible increase in diametral clearance, (mm)
C basic dynamic load rating for radial and angular contact ball or radial
roller bearings, kN (lbf )
Ca basic dynamic load rating for single-row, single- and double-direction
thrust ball or roller bearings, kN (lbf )
Ca1, Ca2; ;
Can
basic load rating per row of a one-direction multi-row thrust ball or
roller bearing, each calculated as single-row bearing with Z1,
Z2; ; Znballs or rollers, respectively
Cn capacity of the needle bearing, kN (lbf )
Co basic static load rating for radial ball or roller bearing, kN (lbf )
Coa basic static load rating for thrust ball or roller bearings, kN (lbf )
d bearing bore diameter, m (in)
db diameter of ball, m (in)
di shaft or outside diameter of inner race used in Eqs (23-246) and
(23-247), m (in)
do inside diameter of outer race of needle bearing, m (in)
dr roller diameter (mean diameter of tapered roller), m (in) diameter of
needle roller, m (in)
d1, d2 diameter of spherical balls or cylindrical rollers used in contact stress
[Eqs (23-250) to (23-253)], m (in)
D outside diameter of bearing, m (in)
D1 diameter of revolving race, m (in)
Dw diameter of ball, mm
E modulus of elasticity, GPa (psi)
f a factor use in Eq (23-155)
fa application factor to compensate for shock continuous duty or
inequality of loading
fc a factor which depends on the geometry of the bearing components, the
accuracy to which the various bearing parts are made and thematerial used in Eqs (23-187), (23-188), and (23-199) to (23-202);
a factor which depends on the units used, the exact geometrical shape of
the load-carrying surfaces of the roller and rings (or washers in case
of thrust bearing), and the accuracy to which the various bearingparts are made and the material, used in Eqs (23-207), (23-208)
fd a factor for the additional forces emanating from the mechanisms
coupled to the gearing used in Eq (23-154)
fk a factor for the additional forces created in the gearing itself used in
Eq (23-154)
fL index of dynamic stressing
fn speed factor for ball bearings according to Table 23-37
speed factor for roller bearings according to Table 23-38
fs index of static bearing
fnt speed factor used in tapered roller bearing
fo a factor used in Eqs (23-161) and (23-167)
foa a factor used in Eqs (23-152) and (23-154)
DESIGN OF BEARINGS AND TRIBOLOGY 23.77
Trang 24theoretical tooth load, kN (lbf )
Fa thrust load, kN (lbf )
Faa applied thrust load, kN (lbf )
Far thrust component of pure radial load F, due to tapered roller,
kN (lbf )
Fbs shaft load due to belt drive, kN (lbf )
Fc static load, kN (lbf )
Fe radial equivalent load from combination of radial and thrust loads or
effective radial load, kN (lbf )
Feffg effective tooth load, kN (lbf )
Fna net thrust load, kN (lbf )
Fnt net thrust load on the tapered roller bearing, kN (lbf )
Fr radial load capacity of ball bearing, kN (lbf )
radial bearing load, kN (lbf )
i number of rows of balls in any one bearing
k constant used in Eqs (23-156), (23-158) to (23-160)
Ka application factor, Eq (23-186)
Kh hardness factor used in Eq (23-247)
Kt life load factor taken from the curve in Fig 23-55 marked
‘‘T-needle’’ and used in Eq (23-247)
Kn a constant used in Eq (23-152) and Eq (23-153)
l length of needle bearing, m (in)
leff the effective length of contact between one roller and that ring (or that
washer in case of thrust bearing) where the contact is the shortest(overall roller length minus roller chamfers or minus grindingundercuts), m (in)
L life of bearing at constant speed, rpm
life of bearing at constant speed, h
life corresponding to desired reliability, R, used in Eq (23-194)
LB10 life factor corresponding to desired B-10 hours of life expectancy used
n1 speed of the inner race, rpm
n2 speed of the outer race, rpm
P equivalent dynamic load, kN (lbf )
Pa equivalent dynamic thrust load, kN (lbf )
Pm mean load, kN (lbf )
Pmax maximum load, kN (lbf )
Pmin minimum load, kN (lbf )
Po static equivalent load, kN (lbf )
Poa static equivalent load for thrust ball or roller bearings under combined
radial and thrust loads, kN (lbf )
qi percentage time of ith speed
R10 0.90 reliability corresponding to rating life
X radial factor used in Eqs (23-177b), (23-182), (23-190), (23-210),
and (23-180)
Trang 25Xo radial factor used in Eqs (23-162), (23-165), (23-173c) and (23-157)
Tables (23-37), (23-38), (23-39)
Y thrust factor used in Eqs (23-163), (23-166), (23-173), (23-178), and
(23-180)
Yo thrust factor used in Eqs (23-162), (23-165), and (23-157)
number of balls carrying thrust in one direction
number of rollers per row
number of rollers carrying thrust in single-row one-direction bearing
number of needle-rollers
Z1, Z2; ; Zn number of balls or rollers in respective rows of one-direction multi-row
bearings
nominal angle of contact, that is, nominal angle between the line of
action of the ball load and a plane perpendicular to the bearing axisthe angle of contact, that is, the angle between the line of action of the
roller resultant load, and a plane perpendicular to the bearing axis
cðmaxÞ maximum compressive stress, MPa (psi)
max maximum shear stress, MPa (psi)
DESIGN OF BEARINGS AND TRIBOLOGY 23.79
... used in Eqs ( 23 - 1 62) , ( 23 - 165), ( 23 - 173c) and ( 23 - 157)Tables ( 23 - 37), ( 23 - 38), ( 23 - 39)
Y thrust factor used in Eqs ( 23 - 1 63) , ( 23 - 166), ( 23 - 1 73) , ( 23 - 178), and
( 23 - 180)
Yo... factor used in Eqs ( 23 - 177b), ( 23 - 1 82) , ( 23 - 190), ( 23 - 21 0),
and ( 23 - 180)
Trang 25Xo...2< /small> =2? ??tan 1 =2? ??
W ẳ24 Q
3< /small>d2< /small>
2? ?? 3e e3< /small>1 e2< /small>ị2< /small>
ð 23 - 148Þ