How Unbalance Affects Bearing Life
By Ahmed M. Al-Abdan, Maintenance Engr, GED/MD, Saudi Electricity Company
Mass unbalance in a rotating system often produces excessive synchronous forces that reduce the life span of
various mechanical elements. First of all we will define unbalance case. Unbalance is basically very simple
case caused by an asymmetry in the rotating element that results in an offset between the shaft centerline and
center of mass (see Figure 1). Asymmetry can be an off-center weight distribution, or it can be a thermal
mechanism that produces uneven heating and bowing of the rotor, or it can be an electrical effect that
produces uneven magnetic field.
Figure 1
There are many purposes of balancing done to a rotating element. One of that purposes is to reduce the force
at the bearings. To eliminate the mass unbalance we have to requalize mass distribution of rotating element
around its centerline by using add or remove weights from available planes, which will reduce the centrifugal
force at the bearings. This can be seen from the following formula:
Where:
Fc:
Centrifugal force at bearings (pounds)
u:
Unbalance weight (ounces)
r:
Radius of unbalance weight (in)
For example, if an unbalance weight of 1.25 ounces is added to a rotor at a 6 inch radius, the resulting force
for a rotor turning at 2000 rpm.
An important question to consider is how does this extra 53.1 pounds of force from unbalance affect the life of
the bearing? To answer this question, we should define a problem can affect the life of bearings which called
fatigue life, many of bearing manufacturer introduce their way to calculate fatigue life i.e. skf by using the
following formula.
Where
a:
a factor for operating condition (Usually between 1.5 and 2.0 Used 1.8)
c:
rated bearing load (7460 Ibs for this Example)
P:
actual radial load (395 Ibs for this Example)
N:
rpm of the machine
The calculated L10 life is 101000 hours or about 11.5 years. If the additional unbalance load of 53 pound is
added to the radial load (p) of 395 pounds the calculated L10 is reduced to 69253 hours or 7.9 years. The
additional 53 pounds of force due to unbalance reduce life by 30%.
Calculate Bearing Life
Timken
Posted 9-13-03
Basis for Calculation | Bearing Life Equation | Bearing Ratings | L10Life Calculation
Basis for calculation
Bearing life is defined as the length of time, or the number of revolutions, until a fatigue spall of a specific size
develops. This spall size, regardless of the size of the bearing, is defined by an area of 0.01 inch 2 (6 mm2). This
life depends on many different factors such as loading, speed, lubrication, fitting, setting, operating temperature,
contamination, maintenance, plus many other environmental factors. Due to all these factors, the life of an
individual bearing is impossible to predict precisely. Also, bearings that may appear to be identical can exhibit
considerable life scatter when tested under identical conditions. Remember also that statistically the life of
multiple rows will always be less then the life of any given row in the system. For bearings where it is impossible
to test a large number of bearings, the long experience of The Timken Company will help you in your bearing life
calculation.
L10 life
L10 life is the life that 90 percent of a group of apparently identical bearings will complete or exceed before the
area of spalling reaches the defined 0.01 inch2 (6 mm2) size criterion. If handled, mounted, maintained,
lubricated and used in the right way, the life of your tapered roller bearing will normally reach and even exceed
the calculated L10 life.
If a sample of apparently identical bearings is run under specific laboratory conditions, 90 percent of these
bearings can be expected to exhibit lives greater than the rated life. Then, only 10 percent of the bearings tested
would have lives less than this rated life. Figure 3-48 shows bearing life scatter following a Weibull distribution
function with a dispersion parameter equal to 1.5.
Bearing life equation
As you will see it in the following, there is more than just one bearing life calculation method, but in all cases the
bearing life equation is :
L10 = (C / P)10/3 (B / n) a
L10 in hours
C = radial rating of the bearing in lbf or N
P = radial load or dynamic equivalent radial load applied on the bearing in lbf or N. The calculation of P depends
on the method (ISO or Timken) with combined axial and radial loading
B = factor dependent on the method ; B = 1.5 106 for the Timken method (3000 hours at 500 rev/min) and
106/60 for the ISO method
a = life adjustment factor ; a = 1, when environmental conditions are not considered ;
n = rotational speed in rev/min.
This can be illustrated as follows :
Doubling load reduces life to one tenth. Reducing load by one half increases life by ten,
Doubling speed reduces life by one half. Reducing speed by one half doubles life.
In fact, the different life calculation methods applied (ISO 281, Timken method...) differ by the selection of the
parameters used (i.e. the Timken formula is based on 90 million revolutions, whereas the others are based on 1
million revolutions).
Bearing ratings
Depending on the life calculation method used, the bearing ratings have to be selected accordingly. The "C r"
rating, based on one million revolutions, is used for the ISO method, and the "C 90" rating, based on 90 million
revolutions, is utilized for the Timken method.
The Timken rating is also published based on 1 million revolutions : C1 = C90 3.857
This will enable you to make a direct comparison between Timken bearings and those using ratings evaluated on
a basis of 1 million revolutions. However, a direct comparison between ratings of various manufacturers can be
misleading due to differences in rating philosophy, material, manufacturing and design. In order to make a true
geometrical comparison between the ratings of different bearing suppliers, only the rating defined following the
ISO 281 equation should be used. However, by doing this, you do not take into account the different steel
qualities from one supplier to another.
ISO 281 Dynamic Radial Load Rating Cr
This bearing rating equation is published by the International Organization for Standardization (ISO) and AFBMA.
These ratings are not published by The Timken Company nor by any other bearing manufacturers. However, they
can be obtained by contacting our company.
The basic dynamic load rating is function of:
Cr = bm fc (i Lwe cos a)7/9 Z3/4
Dwe29/27
Cr = radial rating
bm = material constant (ISO 281 latest issue
specifies a factor of 1.1)
fc = geometry dependent factor
i = number of bearing rows within the
assembly
Lwe = effective roller contact length
a = bearing half-included outer race angle
Z = number of rollers per bearing row
Dwe = mean roller diameter
Timken Dynamic Radial Load Rating C90
Even though the ISO method allows you to compare different bearing suppliers, the basic philosophy of The
Timken Company is to provide you with the most practical bearing rating for your bearing selection process. Since
1915 The Timken Company has developed and validated a specific rating method for its tapered roller bearings.
The published Timken C90 ratings are based on a basic rated life of 90 million revolutions or 3000 hours at 500
rev/min.
To assure consistent quality worldwide, we conduct extensive bearing fatigue life tests in our laboratories. These
audit tests result in a high level of confidence in our ratings. The basic dynamic load rating is used to estimate
the life of a rotating bearing and is a function of:
C90 = M H (i x Leff cos a)4/5 Z7/10
Dwe16/15
C90 = radial rating
M = material constant
H = geometry dependent factor
i = number of bearing rows within the
assembly
Leff = effective roller contact length
a = bearing half-included outer race angle
Z = number of rollers per bearing row
Dwe = mean roller diameter
A rating based on 90 million revolutions is more realistic as most applications equal or exceed this duration. For
double row bearings in which both rows are loaded equally, the two-row rating considers the system life of the
assembly as follows:
C90(2) = 24/5 C90 or C90(2) = 1.74 C90
The basic radial load rating of a four-row assembly is taken as two times the double row rating :
C90(4) = 2 C90(2)
and for a six-row assembly as three times the double row rating :
C90(6) = 3 C90(2)
The Timken Company also publishes K factors for its bearings. This factor is the ratio of basic dynamic radial load
rating to basic dynamic thrust load rating of a single row bearing:
The Timken Company also publishes K factors for its bearings. This factor is the ratio of basic dynamic radial load
rating to basic dynamic thrust load rating of a single row bearing:
K = C90 / Ca90
The smaller the K factor, the steeper the bearing cup
angle (fig. 3-51). The relationship can also be
geometrically expressed as:
K = 0.389 x cot a
a = half included outer race angle
L10 life calculation
Single row bearing
Tapered roller bearings are ideally suited to carry all types of
loads : radial, axial or any combination. Due to the tapered
design of the bearing, a radial load will induce an axial reaction
within the bearing which must be equally opposed to avoid
separation of the inner and outer rings. The ratio of the radial to
the axial load (external axial load and induced load), the setting
and the bearing included cup angle determine the load zone in a
given bearing. This load zone is defined by an angle which
delimits the rollers carrying the load. If all the rollers are in
contact and carry the load, the load zone is referred to as being
360 degrees.
In the case of combined loads, a dynamic equivalent radial load
must be calculated to determine bearing life. The equations
presented below give close approximations of the dynamic
equivalent radial loads. More exact calculations using computer
programs can be made that take into account such parameters
as bearing spring rate, setting and supporting housing stiffness.
Combined radial and thrust load
ISO Method
Thrust Condition
Thrust Condition
Net Bearing Thrust Load
Net Bearing Thrust Load
Dynamic Equivalent Radial Load Dynamic Equivalent Radial Load
Bearing A
Bearing A
PA = FrA
Bearing B
Bearing B
PB = FrB
L10 Life
Timken Method
Thrust Condition
Thrust Condition
Net Bearing Thrust Load
Net Bearing Thrust Load
Dynamic Equivalent Radial Load Dynamic Equivalent Radial Load
Bearing A
Bearing A
PA = 0.4FrA + KAFaA
PA = FrA
if PA < FrA, PA = FrA
Bearing B
Bearing B
PB = 0.4FrB + KBFaB
PB = FrB
if PB < FrB, PB = FrB
L10 Life
ISO 281 Factors
e = 1.5 tan a Y = 0.4 cot a Y 1 = 0.45 cot a Y 2 = 0.67 cot a
Two-Row Bearing
Thrust Load Only
ISO Method
Thrust Condition
FaA = Fae
FaB = 0
Thrust Load
FaA = Fae
Dynamic Equivalent Load FaB = 0
PA = YAFaA
PB = 0
L10 Life
Timken Method
Thrust Condition
FaA = Fae
FaB = 0
L10 Life
Thrust Load
FaA = Fae
FaB = 0
ISO Method
Thrust Condition
Thrust Condition
Dynamic Equivalent Radial Load
PAB = FrAB + Y1ABFae
PC = FrC
Dynamic Equivalent Radial Load
PAB = 0.67FrAB + Y2ABFae
PC = FrC
L10 Life
Timken Method
Thrust Condition Dynamic Equivalent Radial Load
PA = 0.4FrAB + KAFae
PB = 0
PC = FrC
Thrust Condition Dynamic Equivalent Radial Load
PA = 0.5FrAB + 0.83KAFae
PB = 0.5FrAB - 0.83KAFa
PC = FrC
L10 Life
