Discussion
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
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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