design of rolling contact bearings
Rolling Contact Bearings (Antifriction Bearings)
Static Equivalent load [Po]
The static equivalent load may be defined as the static radial load or axial load which if applied would cause the same total permanent deformation at the most heavily stressed ball and race contact as that which occurs under the actual condition of loading [is under combined radial & axial or thrust loads)
Statically Loaded Bearings. The bearings which do not rotate or rotate at a very small speed (n < 1 rev/, min) are called as statically loaded. An example of a 'stationary radial load', one that does not rotate, acts in a constant direction is the belt load. The limiting load of such bearings is determined by the magnitude of the permanent deformation of the contact surfaces, and not by the service life of the bearing parts. The earliest* work on rolling element bearings was carried out by Hertz and Striebeck. The maximum load which would be carried by a ball is given by the formula :
where Fr = radial load on the bearings
z = number of balls in the bearing
In this formula, it has been assumed that there is no radial clearance between balls and races. Radial clearance has marked effect on the load distribution between balls and on the magnitude of P0. For this reason, the above equation is modified as:
The basis static load rating “C0” which corresponds to a total permanent deformation of ball and race, at the most heavily stressed contact point, equal to 0.0001 times the ball diameter, is given by the following formula:
where i = number of rows of balls in any one bearing.
z = number of balls per row
D = Ball diameter
a = Angle between the line of action of ball load and a plane normal to bearing axis,
f0 = a factor. In (N.m) units, its values are:
= 3.34 × 106 for self aligning ball bearings.
= 12.26 × 106 for radial contact and angular contact ball bearings.
If the bearing is under the combined radial and thrust loads, the static equivalent load P0 shall be the greatest of those obtained from the following formulas:
where
Y0 = a thrust factor
Fr = radial load
Fa = thrust load.
The values of X0 and Y0 are given in Table.
The equivalent load is defined as that hypothetical load which, if applied, will have the same effect on bearing life as the actual loads.
Bearing Type
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Single row bearing
|
Double row bearing
| ||
X0
|
Y0
|
X0
|
Y0
| |
Radial contact groove ball bearings
|
0.6
|
0.5
|
0.6
|
0.5
|
a = 20o
|
0.5
|
0.42
|
1
|
0.84
|
Angular contact = 25o
|
0.5
|
0.38
|
1
|
0.76
|
grooove ball = 30o
|
0.5
|
0.33
|
1
|
0.66
|
bearings = 35o
|
0.5
|
0.29
|
1
|
0.58
|
= 40o
|
0.5
|
0.26
|
1
|
0.52
|
Self aligning ball bearings
|
0.5
|
0.22 cota
|
1
|
0.44 cot
|
Dynamic Load Carrying Capacity of a bearing is defined as the radial load in radial bearings (or thrust load in thrust bearings) that can be carried for a minimum life in this definition is the life which 90% of the bearings will reach or exceed before fatigue failure.
Dynamically loaded Bearings. Equivalent Dynamic Load is defined as the constant radial load in radial bearings (or thrust load in thrust bearings) which if applied to the bearing would give same life as that which the bearing will attain under actual condition of forces. The equivalent load for a dynamically loaded bearing under the combined section of radial and thrust loads is always referred as a radial load and is given by the formula:
P = XVFr + YFa (32)
P = VFr, whichever is larger
Value for X0 and Y0 (IS3823) (Part I) – 1966]
where V = Rotation factor
= 1.0 for all types of bearings if inner race is rotating.
= 1.2 for all types of bearings except self aligning, if inner race is stationary.
= 1.0 for self aligning for rotation of either race.
The value for the radial factor “X” and thrust factor “Y” for most of the bearings can be assumed as given below:
X = 1 : Y = 1.5
For angular contact bearings, X = 0.50; Y = 1.0
The self aligning bearing:
x = 0.50 : y = 2.50
Life:
The life of an individual ball (or roller) bearing may be defined as the number of revolutions (or hrs at some given constant speeds) Which the bearing runs before the first evidence of fatigue crack develops in the malarial of one of one of the rings or any of the rolling dements.
It conditions of friction, noise and smoothness are not critical, a much higher premier deformation can be tolerated and consequently static loads up to 4 times the static load carrying capacity may be permissible. It extreme smoothness of operation is desired, a smaller permanent deformation is desired.
Rating life of a group of apparently is defined as the number of revolutions (or hrs at some given constant speed) that 90% of a group of bearings will complete or exceed before the first evidence of faligue develops (ie only 10% of a group of bearings fail due to faligue)
For majority of bearings the actual life is more than the rating life.
Life of Bearing
Since life of a single bearing is difficult predict it is defined as an average of a group of bearings.
Recommended Bearing Life
Type of Application
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Life, hrs.
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Instruments and apparatus for infrequent use.
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Upto 500
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Aircraft engines
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500 – 2000
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Machines for short or intermittent service where service interruption is of minor importance
|
4000 - 8000
|
Machines for intermittent service where reliable operation is of great importance
|
8000 - 14000
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Machines for 8 – hr. service which are not always fully utilized
|
14000 – 20000
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Machines for continuous 24-hr, service
|
50000 - 60000
|
Machines of continuous 24-hr, service where reliability is of extreme importance
|
100000 - 200000
|
Reliability
Reliability of a Bearing [R] is defined as the ratio of the no. of bearing which have successfully completed L millions revolutions to the total no of bearings under test.
Relation bearing Life and reliability
b = 1.17
R = Reliability
L = Life of bearing in revolution
If L90 is the life of the bearing corresponding to a reliability of 90% [ie R90]
L is the life of the bearing corresponding to any reliability R.
then
Selection of radial ball bearing
The selection of a rolling element bearing concerns only the selection of the most suitable bearing from the catalogue of a manufacture, according to the procedure given in the catalogue.
The following procedure can be followed:
1. The equivalent load coming on the bearing is calculated by the help of eqn. (17.32). This equation does not account for any shock or impact forces and temperature conditions which a bearing may experience. Incorporating these two factors, the design load F, is calculated from the equivalent load, P, as follows :
F = P × Ka × KT
where Ka = application factor or service factor, Table 17,21.
Kt = temperature factor. Table 17,22.
2. Now it is a usual practice with bearing manufacturers to specify the rated radial bearing load capacity to a certain speed in rev/min, and a certain life in hours (L,0 or Average). For example, in Tables 17.23 and" 17.27, the radial load ratings are at 10,000 hours of average life at 500 rev/min. The actual or design radial load, F, may have to be carried for a different life and at a different speed than given in the catalogues or Tables • 17.23 and 17.27. It is, therefore, important that when comparing the rated capacity of bearings made by different manufacturers, capacities are all reduced to the same life expectancy and speed. We can find the required radial load rating, CR, for these conditions with the help of equation (17.26) that is,
where
Lc = catalogue life of bearing, hrs.
nd = Rotational speed of bearing, rev/min.
nc = Catalogue rotational speed, rev/min.
or
where
KS = Speed factor =
Values for Ka
Type of Service
|
Ka
| |
Ball Bearings
|
Roller Bearings
| |
Uniform and Steady Load
|
1.0
|
1.0
|
Light shock load
|
1.5
|
1.0
|
Moderate shock load
|
2.0
|
1.3
|
Heavy shock load
|
2.5
|
1.7
|
Extreme shock load
|
3.0
|
2.0
|
Values of Kt.
toC =
|
125
|
150
|
175
|
200
|
225
|
250
|
Kt =
|
1.05
|
1.10
|
1.15
|
1.25
|
1.35
|
1.40
|
3. After determining the required radial load, the suitable bearing is selected from Table 17.23 so that the radial load capacity given in the table is more than the required radial load.Thrust Ball Bearings
The three main types of thrust ball bearings are:
(a) Singe direction thrust bearings with flat seats
(b) One dissection thrust bearings with grooved seats,
(c) Double direction thrust bearings with grooved seats.
Assuming that only 80 percent of the balls take the axial load, the force acting on each ball will be given as:
The basic static load rating of the thrust ball bearing is given by the following formula:
where f0a = constant , 49.05 × 106 per units of (N.m)
z = number of balls carrying thrust in one direction
D = Ball diameter
a = normal angle of contact.
The static equivalent load for thrust ball bearings with contact angle
For a pure thrust ball bearing under dynamic loading, in equation (32), the equivalent load is simply equal to Fa. For the selection of a thrust ball bearing, the same procedure can be adopted as for radial ball bearing. The rated thrust capacity may be taken as 75 percent of the radial capacity as given in Table 23.
References
1. Mechanical Engineering Design – Joseph Shigley
2. Machine Design – Mubeen
3. Machine Design – Black
4. Principles of Lubrication – Cameron A.
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