Miles Equation Worked Interpretation

What Is It?

This article applies Miles equation to illustrative cases to show how each parameter — natural frequency, damping and PSD magnitude — affects the estimated RMS response. The goal is to develop engineering intuition for the sensitivity of the result, not to draw universal conclusions that may not apply to specific cases.

Why It Matters

Understanding the sensitivity of the Miles equation result to each parameter helps engineers identify which parameters most strongly control the response and where design effort should be focused. It also helps in interpreting full analysis results — if the full analysis disagrees with Miles, understanding the parameter sensitivities helps diagnose why.

Effect of Natural Frequency

The natural frequency enters Miles equation in two places: directly as fn and through the input PSD value G(fn). If the input PSD is flat, the RMS acceleration is proportional to √fn — higher natural frequencies produce higher response. However, real PSDs are not flat — the PSD value at fn may increase or decrease with frequency, and this effect often dominates.

Miles equation:

a_rms  =  √( π · f_n · G_in(f_n) / (4 · ζ) )

If G_in is flat (constant):
  a_rms  ∝  √f_n    (higher fn → higher response)

If G_in decreases with frequency:
  The G(fn) term may dominate —
  higher fn may produce LOWER response

Conclusion: the effect of fn depends on the
PSD shape. No universal rule.

The effect of natural frequency on the Miles response depends on the PSD shape. No universal conclusion can be drawn — the PSD value at fn is as important as fn itself.

Effect of Damping

Damping enters Miles equation as 1/√ζ in the RMS and as 1/ζ in the response PSD. The sensitivity is strong: halving the damping increases the RMS by √2 ≈ 1.41 and the peak response PSD by a factor of 4. This makes damping the most influential parameter in the Miles equation — and the one with the greatest uncertainty.

Damping sensitivity:

a_rms  ∝  1/√ζ

Damping change:  ζ → ζ/2
RMS change:  a_rms → a_rms × √2  ≈  1.41×
Peak PSD change:  |H|² → |H|² × 4

Illustrative:
  ζ = 5%:  a_rms = √(π · fn · G(fn) / 0.20)
  ζ = 2.5%: a_rms = √(π · fn · G(fn) / 0.10) = √2 × above
  ζ = 1%:  a_rms = √(π · fn · G(fn) / 0.04) = √5 × above

Effect of PSD Magnitude

The input PSD magnitude enters linearly under the square root: a_rms ∝ √G(fn). Doubling the PSD increases the RMS by √2. This is a weaker sensitivity than damping (which is 1/√ζ) but still significant. The PSD magnitude is typically the best-known parameter — it comes from the specification or measurement — so its uncertainty is usually less than damping uncertainty.

PSD sensitivity:

a_rms  ∝  √G_in(f_n)

PSD change:  G → 2G
RMS change:  a_rms → a_rms × √2  ≈  1.41×

PSD change:  G → 4G
RMS change:  a_rms → a_rms × 2

Combined Sensitivity

The combined sensitivity to all three parameters can be substantial. A factor of 2 uncertainty in damping, combined with a factor of 2 uncertainty in the PSD level, produces a factor of 2 uncertainty in the RMS response. For vibration fatigue, where damage depends on stress to a power, this translates to a much larger uncertainty in life. This is why sensitivity studies are essential.

Illustrative Cases

The following illustrative cases show the Miles equation applied with different parameter values. The values are illustrative — they demonstrate the sensitivity, not specific engineering conclusions.

Illustrative cases (flat PSD = 0.04 g²/Hz):

Case 1:  fn = 100 Hz, ζ = 5%
  a_rms = √(π × 100 × 0.04 / 0.20) = √(62.8) = 7.93 g

Case 2:  fn = 100 Hz, ζ = 2.5%
  a_rms = √(π × 100 × 0.04 / 0.10) = √(125.7) = 11.2 g
  (×1.41 from halving damping)

Case 3:  fn = 200 Hz, ζ = 5%
  a_rms = √(π × 200 × 0.04 / 0.20) = √(125.7) = 11.2 g
  (×1.41 from doubling fn with flat PSD)

Case 4:  fn = 100 Hz, ζ = 5%, G = 0.08 g²/Hz
  a_rms = √(π × 100 × 0.08 / 0.20) = √(125.7) = 11.2 g
  (×1.41 from doubling PSD)

Note: all values are illustrative.

No Universal Conclusions

The sensitivities shown above should not be generalised into universal rules. The actual sensitivity in a specific case depends on the PSD shape, the number of contributing modes, the frequency range and the structure. The value of this interpretation is in developing engineering intuition, not in producing rules of thumb that may not apply.

These sensitivities are illustrative. Do not generalise into universal rules. The actual sensitivity depends on the PSD shape, mode count and structure. Always evaluate sensitivity for the specific case.

Key Takeaways

• Damping has the strongest sensitivity: a_rms ∝ 1/√ζ — halving damping increases RMS by √2
• PSD magnitude: a_rms ∝ √G — doubling PSD increases RMS by √2
• Natural frequency effect depends on PSD shape — no universal rule
• Combined uncertainty in damping and PSD can produce factor-of-2 uncertainty in RMS
• These sensitivities are illustrative — do not generalise into universal rules