MEC-37

Miner linear damage

D = Σ ni/Ni. Failure is predicted at D = 1 under block loading.

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FatiguePalmgren–Miner

Governing equation

D=\sum_i n_i/N_i

$n_1 \in \left[100, 5\times 10^{6}\right]$

where

n_1: Cycles block 1 (—)
N_1: Life block 1 (—)
n_2: Cycles block 2 (—)
N_2: Life block 2 (—)
D: Damage sum (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

D=\sum_i n_i/N_i

$n_1 \in \left[100, 5\times 10^{6}\right]$

$n_1 = 100$ $D = 0.161$

Along the curve

②

Domain of the variable — blue

n_1_{\min} = 100,\quad n_1_{\max} = 5\times 10^{6}

$n_1 \in \left[100, 5\times 10^{6}\right]$

$n_1 = 100$ $n_1 = 100$

Along the curve

③

Response curve — green

D=\sum_i n_i/N_i

$n_1 \in \left[100, 5\times 10^{6}\right]$

$n_1 = 100$ $D = 0.161$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}D}{\mathrm{d}n_1}

$n_1 \in \left[100, 5\times 10^{6}\right]$

$n_1 = 31545.9$ $D^{\prime} = 5\times 10^{-6}$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-37 — Miner linear damage) is the form associated with Palmgren–Miner. Working symbols: $n_1$ , $N_1$ , $n_2$ , $N_2$ $\rightarrow$ $D$ . Palmgren and Miner assumed each cycle spends a fraction 1/Ni of life, independent of sequence.

Purpose

Purpose: compute

D

from

n_1

N_1

n_2

N_2

in Mechanical via

D=\sum_i n_i/N_i

D = Σ ni/Ni. Failure is predicted at D = 1 under block loading. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n_1 = 20000.000\,\mathrm{—}

N_1 = 200000.000\,\mathrm{—}

n_2 = 80000.000\,\mathrm{—}

N_2 = 500000.000\,\mathrm{—}

, the governing relation

D=\sum_i n_i/N_i

yields

D = 0.260\,\mathrm{—}

. Two blocks, two life bars, a damage sum. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Cycles block 1 n_120000 —Life block 1 N_1200000 —Cycles block 2 n_280000 —Life block 2 N_2500000 —

Outputs

Damage sum D0.260 —

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Narration of this film

Two blocks, two life bars, a damage sum.

Palmgren and Miner assumed each cycle spends a fraction 1/Ni of life, independent of sequence.

Reading speed

Watch on YouTube