Palmgren-Miner equation derivation

[k.udhay]

TL;DR

This equation is used in ISO standard ISO 6336-6. I want to understand how it is derived.

Summary: This equation is used in ISO standard ISO 6336-6. I want to understand how it is derived.

This equation helps to consolidate damage caused by multiple torque, cycle bins. A bin means a torque applied for a given number of cycle. As an outcome of the equation, a single equivalent torque that produces the total damage caused by multiple bins for the total number of cycles as original bins.

The equation is:

Pl. note that coordinates n1,T1; n2,T2 and n3,T3 do not need to be on the same line.

My objective is to derive the above equation. While I attempted to do that, I could progress to this level:

Can someone help to proceed further to reach until the equivalent torque formula given in the ISO standard? Many thanks in advance!

 

[pbuk]

Hi, I was about to answer this and suggest it would be better in the Engineering topic here (as this is all about engineering approximation rather than scientific derivation), but while searching for refs I realized that you have posted this on another forum where it has been answered: https://www.eng-tips.com/viewthread.cfm?qid=460402

 

[berkeman]

(and thread title corrected: "Palgmer-Miner" --> "Palmgren-Miner")

https://en.wikipedia.org/wiki/Fatigue_(material)#Miner's_rule