13.1 Calculating Lifetime according Miners rule

The Palmgren-Miner Linear-cumulative-fatigue-damage-theory (Miner’s Rule) is used to calculate the resultant pitting or bending fatigue lives for gears that are subjected to loads which are not of constant magnitude but vary over a wide range. According to Miner’s Rule, failure occurs when:

n1    n2        ni
N--+ N-- + ...+  N--=  1
 1     2          i


ni =number of cycles at the ith stress level.
Ni =number of cycles to failure correspontiing to the ith stress level.
ni∕Ni=damage ratio at the ith stress level.

Instead of load cycles we can alo use lifetimes:

l1-   l2-       li-
L1 +  L2 + ...+  Li = 1


li =time at a the ith stress level.
Li =permissible lifetime at the ith stress level.
li∕Li=damage ratio at the ith stress level.

Assuming the fraction of time at each stress level is known rather than the actual number of cycles or times, then:

l1  =   α1 ⋅ L
l2  =   α2 ⋅ L

li =   αi ⋅ L


αi=fraction of time at the ith stress level.
L =Resultant number of cycles to failure under the applied load spectrum.

Defining the time ratio as:

αi =  li∕L =  ni∕N

Miner’s Rule may be rewritten as:

   L       L            L
α1--- + α2--- + ...+ αi ---= 1
  L1      L2            Li

Which may be solved for the resultant life:

L = ---------1--------
     αL11 + αL22 + ...+ αLi

The load spectrum is defined by the time ratio, αi, and the load ratio, βi and additionally a speed ratio ωi is needed for the calculation of the permissible lifetimes Li.


βi=instantaneous load/baseline load
ωi=instantaneous speed/nominal load

The baseline load is entered with the Load Data input screen by specifying the transmitted horsepower and speed of the pinion. The load spectrum is entered on the page Lifetime: