Total System Life Adjustment Factors for Reliability

The formulas below are used to generate the total life of a multibearing system. Many bearing manufacturers give L10 life guidelines and fail to qualify whether it applies to a single bearing or a multibearing system life.  This analysis is only relevant when bearings interact in a single component.  Also, when complete, components interact with each other to form a machine.

1/L (Total) = 1/L1 + 1/L2 + 1/ L3 + 1/L4 + ... + 1/Ln

(Miner’s - Equation - Gaussian Distribution)

The above equation is Gaussian and slightly different from the theoretically correct Weibull distribution which is normally applied to bearings.  However, the difference is minor and is conservative.  The Weibull Distribution is generally not worth the added mathematics involved.  It only becomes relevant when very large bearing systems are involved.

The totally correct Weibull formula is as follows:

L (Total) = [ (1/L1)^e +(1/L2)^e + (1/L3)^e + - - - + (L/n)^e]^ - 1/e

(Weibull Distribution)

Where >>>

L(Total) = This is the total component or system life.
L1 …+ Ln = These are individual element life’s or system Lxx life.
e = 10/9 for ball bearings
e = 9/8 for roller bearings

In theory, if the mathematical model from above would assume that each bearing has the same L10 life. Then a 2- bearing system, such as a spindle would have one-half (½) of the individual bearing L10 life in theory.  When a 4- bearing system is used, such as a gearbox, the bearing L10 life is approximately one-fourth (¼) of the individual bearing L10 life in theory.  Also this applies only to a Gaussian method or Weibull method.  However, in reality the L10 life in components is almost never equal.  Therefore, the above example is used only to illustrate the effects of interaction of a multibearing system.        

Source – SKF, Fafnir, and MPB bearing Corporations.

When the inverse life of the Multibearing system is desired then the following is used. >>>

Ln Multiplier = (Number of bearings in the system) ^1/e

(Weibull Distribution)

Where >>>

e = 10/9 for Ball Bearings
e = 9/8 for Roller Bearings

Therefore, if there are two (2) ball bearings in the system then 2^0.9 is 1.866.  Then the capacity of each bearing must be multiplied be 1.866 in order to have a final result of a factor of one (1) which is the base value of the multibearing system.