Fatigue From Vibration⚠ unverified

Understanding the Parameters

• RMS stress $\sigma_{rms}$ — the statistical severity of the random stress, from the response PSD (Random Vibration Rms). Because it enters as $\sigma_{rms}^m$ with $m \approx 3$–$10$, damage is extremely sensitive to it: a 25 % higher RMS stress with $m = 5$ triples the damage.

• S-N exponent $m$ — the (negative-reciprocal) slope of the material's S-N curve on log–log axes; steeper curves (larger $m$) make life fall faster with stress. It is the dominant sensitivity.

• S-N intercept $A$ — the fatigue-strength constant in $N = A/\sigma^m$ (cycles to failure at stress $\sigma$). It must carry $\text{Pa}^m$ units for the damage to be a pure number.

• Exposure time $T$ — damage accumulates linearly with time (Miner's rule): double the exposure, double the damage. Combined with the cycle rate it becomes a cycle count.

• Damage $D$ — a Miner's-rule sum: $D < 1$ safe, $D = 1$ nominal failure, though scatter means design targets $D \ll 1$.

Derivation (Approaching a Proof)

Fatigue damage under variable amplitude uses Miner's rule: each cycle at stress $\sigma$ consumes a fraction $1/N(\sigma)$ of life, where the S-N curve gives $N(\sigma) = A/\sigma^m$. Summing over all cycles,

$$D = \sum_i \frac{n_i}{N(\sigma_i)} = \frac{1}{A}\sum_i n_i\,\sigma_i^{\,m}.$$

For a narrow-band Gaussian process the peak stresses follow a Rayleigh distribution with scale set by $\sigma_{rms}$, and the number of cycles is the rate $\nu$ times time $T$. Taking the expected value of $\sigma^m$ over the Rayleigh distribution gives the rigorous form $D = \nu T (\sqrt2\,\sigma_{rms})^m \Gamma(1+m/2)/A$. The registry **simplifies** this to $\sigma_{rms}^m$ times $T$ over $A$:

$$D \approx \frac{\sigma_{rms}^{\,m}\,T}{A},$$

folding the cycle rate and the distribution factor into the constant. It preserves the crucial $\sigma_{rms}^m$ and linear-$T$ dependences while dropping the exact numerical factor.

Dimensional check. With $A$ in $\text{Pa}^m$: $D = \dfrac{\sigma_{rms}^{\,m}\,T}{A} = \dfrac{\text{Pa}^m\cdot\text{s}}{\text{Pa}^m}$... the time carries a hidden cycle-rate (1/s) in the constant, leaving $D$ dimensionless — a damage fraction, as required. (The registry's dimensionless label for $A$ is the source of the unit inconsistency the note flags.)

History and Development

Random-vibration fatigue rests on Miner's linear damage rule (1945) combined with the spectral (frequency-domain) fatigue methods developed for aerospace from the 1960s. The narrow-band (Rayleigh) approximation was the first tractable model; broadband spectra use rainflow counting or the Dirlik amplitude formula. It is central to qualifying avionics, spacecraft, and vehicle electronics against random vibration environments, where classical constant-amplitude fatigue does not apply.

Related Concepts: Random Vibration Rms, S-N Curve, Fatigue Life cycles, Fatigue Stress Concentration, Modified Goodman Factor, Quality Factor Q

Notes: Simplified narrow-band estimate — exact form adds a Rayleigh $\Gamma(1+m/2)$ factor and cycle rate. $A$ must be Pa$^m$ (registry labels dimensionless). Damage $\propto\sigma_{rms}^m$ (very sensitive, $m\approx3$–$10$) and linear in $T$. $D\ge1$ = failure; use rainflow/Dirlik for design.