Regarding the two classical longitudinal dynamic modes of motion for a conventional fixed-wing airplane, which of the following statements are TRUE?

(A) The short-period mode is characterized by rapid variations in angle of attack ($\alpha$) and pitch rate ($q$) at nearly constant forward airspeed.

(B) The phugoid mode is characterized by large, slow exchanges between kinetic energy (airspeed) and potential energy (altitude) at nearly constant angle of attack.

(C) The damping ratio of the short-period mode is typically much higher than the damping ratio of the phugoid mode.

(D) The phugoid mode has a much higher natural frequency than the short-period mode.

Answer Key:

A, B, C

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Short-Period Mode Characteristics:** Heavy aerodynamic damping from the horizontal tail ($C_{mq}$) yields high damping ratio (typically $\zeta_{sp} = 0.5 - 0.8$) and high frequency ($omega_{sp} \sim 2 - 5\text{ rad/s}$). Pitch attitude and $\alpha$ oscillate rapidly with negligible speed variation (Statement A and C are TRUE). 2. **Stage 2: Phugoid Mode Characteristics:** The aircraft trades airspeed for height like a roller-coaster. The angle of attack remains virtually unchanged ($\Delta \alpha \approx 0$) while $V$ and $h$ oscillate slowly (Statement B is TRUE). 3. **Stage 3: Frequency Comparison:** $\omega_{ph} \approx \sqrt{2} \frac{g}{V_0} \ll \omega_{sp}$. The phugoid has a much lower frequency (periods of 30–120 seconds), making Statement D FALSE. 4. **Stage 4: Official Reference:** Robert C. Nelson, *Flight Stability and Automatic Control*, Chapter 5.

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