How to Use Miles Equation
A procedure for applying Miles equation to estimate the RMS response of an SDOF system under random excitation — when it is valid, how to apply it, and how to interpret the result.
When to use this guide
Use this guide for quick screening estimates of the RMS response at a resonance, before running a full random vibration analysis.
Required inputs
- Natural frequency fn [Hz]
- Damping ratio ζ
- Input PSD value at fn: G(fn) [g²/Hz]
Step-by-step method
- Confirm Miles equation is appropriate: lightly damped (ζ < 5%), single dominant mode, PSD approximately flat near fn
- Read the input PSD value at the natural frequency: G(fn)
- Apply the equation: a_rms = √(π × fn × G(fn) / (4 × ζ))
- For displacement: x_rms = a_rms / (2π × fn)²
- For screening peak: a_peak ≈ 3 × a_rms (3σ screening)
- Compare the screening result against the allowable — if close, run full analysis
- Document the estimate and its statistical basis
Checks
- PSD is approximately flat across the resonance bandwidth (2 × ζ × fn)
- Only one dominant mode contributes at this frequency
- Damping is light (ζ < 5%) — for higher damping, use full analysis
- Result is a screening estimate, not a final assessment