Tools · Random vibration
Miles' equation
A quick estimate of how hard a component responds to broadband random vibration, as long as the input spectrum is roughly flat around its natural frequency.
Q = 1/(2ζ). Q = 10 (ζ = 5%) is a common default when damping is unknown.
| Damping ratio ζ | |
| RMS relative displacement | |
| 3σ relative displacement |
The formulas
Grms = √( (π/2) · fn · Q · PSD(fn) )
zrms ≈ Grms · g / (2πfn)²
The (π/2)·fn/Q term is the effective noise bandwidth of the resonance, so the formula is just "input density × bandwidth × amplification²". Peaks in a Gaussian random response are commonly taken at 3σ (3 × RMS), which covers 99.7% of the time.
Assumes a single dominant mode and a flat input PSD around fn. If the PSD slopes steeply near the resonance, or several modes contribute, integrate the full response PSD instead.
Method & sources
What it calculates
- Miles' equation: Grms = √((π/2)·fn·Q·PSD(fn)), the RMS response of a single resonator to white-noise base acceleration.
- Relative displacement z ≈ Grms·g/(2πfn)²; 3σ values = 3 × RMS for a Gaussian response.
Assumptions & limits
- Single dominant mode; input PSD roughly flat across the resonance bandwidth.
- For sloping or multi-mode inputs, integrate the full response PSD instead.
Verified: fn = 100 Hz, Q = 10, 0.04 g²/Hz gives 7.93 Grms, matching Python.