Which of the following statements regarding the uncontrolled classical natural dynamic stability modes of a conventional fixed-wing airplane in straight, level, unaccelerated flight are **CORRECT**?

(A) The short-period mode is characterized by high frequency and heavily damped oscillations with nearly constant forward airspeed

(B) The phugoid mode is characterized by low frequency, very light damping, and significant interchange between kinetic energy (airspeed) and potential energy (altitude)

(C) During the phugoid mode, the pitch angle $(\theta)$ and angle of attack $(\alpha)$ oscillate with equal amplitude in phase

(D) Increasing static margin $(K_n)$ increases the undamped natural frequency of the longitudinal short-period mode

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles Longitudinal dynamic stability divides into two distinct decoupled oscillatory modes: 1. **Short-Period Mode:** Fast pitching oscillation where angle of attack $\alpha$ and pitch rate $q$ oscillate rapidly. Because the oscillation is so rapid, forward speed $u$ has virtually no time to change ($u \approx \text{constant}$). Restoring moment is proportional to pitch stiffness $C_{m_\alpha}$ (which increases with static margin $K_n$), so $\omega_{n,sp} \propto \sqrt{-C_{m_\alpha}} = \sqrt{C_{L\alpha} K_n}$. 2. **Phugoid Mode:** Long-period, lightly damped oscillation characterized by a large-scale exchange between kinetic and gravitational potential energy. Throughout phugoid motion, the angle of attack remains nearly constant ($\Delta \alpha \approx 0$), meaning $\theta \approx \gamma$ (pitch angle closely follows flight path angle). ### 2. Statement-by-Statement Verification - **Statement A:** **CORRECT.** Short-period mode has high natural frequency, good damping, and almost constant airspeed ($u \approx 0$). - **Statement B:** **CORRECT.** Phugoid mode trades kinetic energy ($V$) for potential energy ($z$) over long periods (often 20 to 60+ seconds) with light aerodynamic damping. - **Statement C:** **INCORRECT.** During the phugoid mode, angle of attack is virtually constant ($\Delta \alpha \approx 0$); only the pitch angle $\theta$ and flight path angle $\gamma$ oscillate. - **Statement D:** **CORRECT.** The short-period natural frequency is $\omega_{n,sp} \approx \sqrt{\frac{-C_{m\alpha} q_\infty S c}{I_{yy}}}$. Since $-C_{m\alpha} = C_{L\alpha} K_n$, increasing static margin $K_n$ directly increases $\omega_{n,sp}$. ### 3. Exam Trap & Standard Reference - **Trap Alert:** Many candidates confuse which parameter is invariant in each mode: in short-period, speed $V$ is constant; in phugoid, angle of attack $\alpha$ is constant. - **Standard Reference:** Nelson, R.C., *Flight Stability and Automatic Control*, McGraw-Hill, Chapter 4 (Longitudinal Motion of the Aircraft).

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