Subtopic: Longitudinal Dynamic Modes: Phugoid and Short Period
Question Statement:
The roots obtained by solving the longitudinal characteristic equation of an airplane are given by $\lambda_{1,2} = -0.005 \pm 0.05i$ and $\lambda_{3,4} = -1.2 \pm 2.5i$. These two pairs of complex conjugate roots correspond respectively to:
Options:
(A) Short period mode and Phugoid mode
(B) Phugoid mode and Short period mode
(C) Dutch roll mode and Spiral mode
(D) Spiral mode and Roll subsidence mode
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.42)**
**1. Intuitive Physical Setup & Given Parameters**:
- Longitudinal characteristic roots:
- Pair 1: $\lambda_{1,2} = -0.005 \pm 0.05i$
- Pair 2: $\lambda_{3,4} = -1.2 \pm 2.5i$
**2. Step-by-Step Dynamic Stability Derivation**:
1. **Modal Frequency and Damping Analysis**:
For any oscillatory mode $\lambda = -\zeta \omega_n \pm i \omega_d$:
- Real part represents the decay rate (damping): $\text{Re}(\lambda) = -\zeta \omega_n$.
- Imaginary part represents the damped oscillation frequency: $\text{Im}(\lambda) = \omega_d = \omega_n \sqrt{1 - \zeta^2}$.
2. **Comparing Pair 1 vs Pair 2**:
- **Pair 1 ($\\lambda_{1,2} = -0.005 \pm 0.05i$)**:
- Very small frequency: $\omega_d = 0.05\text{ rad/s}$ (long period: $T = \frac{2\pi}{0.05} \approx 125.7\text{ s}$).
- Very lightly damped: real part $-0.005$ is close to the imaginary axis.
- This slow, long-period exchange of kinetic and potential energy is the classic **Phugoid mode**.
- **Pair 2 ($\\lambda_{3,4} = -1.2 \pm 2.5i$)**:
- High frequency: $\omega_d = 2.5\text{ rad/s}$ (short period: $T = \frac{2\pi}{2.5} \approx 2.5\text{ s}$).
- Heavily damped: real part $-1.2$ gives rapid subsidence in 1-2 seconds.
- This rapid pitching oscillation is the classic **Short Period mode**.
3. Therefore, Pair 1 and Pair 2 correspond respectively to **Phugoid mode and Short period mode**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** (Short period and Phugoid): ❌ Incorrect. Reversed order.
- **(B)** (Phugoid mode and Short period mode): ✅ **CORRECT**. Low frequency = Phugoid; high frequency = Short Period.
- **(C)** (Dutch roll and Spiral): ❌ Incorrect. These are lateral-directional modes, not longitudinal.
- **(D)** (Spiral and Roll subsidence): ❌ Incorrect. Also lateral-directional modes.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: Longitudinal modes are always Phugoid (low $\omega_n$, low $\zeta$) and Short Period (high $\omega_n$, high $\zeta$).
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