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An aeroplane has a Dutch roll with \( \zeta = 0.08 \) and \( \omega_n = 1.5 \ \mathrm{rad/s} \). Against requirements of \( \zeta \geq 0.08 \) and \( \zeta\omega_n \geq 0.15 \ \mathrm{rad/s} \), the aeroplane

  • A.Meets both requirements comfortably
  • B.Fails both requirements
  • C.Meets the product but fails the damping ratio
  • D.Meets the damping ratio but fails the product, at 0.12

Answer: D

The damping ratio is exactly at its limit, but the product is \( 0.08 \times 1.5 = 0.12 \ \mathrm{rad/s} \), short of the 0.15 required, so the aeroplane meets one criterion and fails the other. This is exactly why specifications state both: a low frequency mode can satisfy a damping ratio while still taking too long in absolute time to decay, and it is the product that fixes the decay rate. Raising the damping ratio to 0.10 would clear it.

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