GATE AE 2014 • Q24
NAT • 1 Mark
Flight Mechanics & Space Dynamics
Dynamic Stability
Subtopic: Short Period & Phugoid Longitudinal Modes
Question Statement:
In most airplanes, the Dutch roll mode can be excited by applying
Options:
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(A) a step input to the elevators
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(B) a step input to the rudder
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(C) a sinusoidal input to the aileron
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(D) an impulse input to the elevators
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2014 Q.24)**
**1. Governing Concept & Theory**:
Governed by principles of **Dynamic Stability** (Aircraft Performance & Stability, 1 Mark, NAT).
**2. Step-by-Step Analytical Derivation**:
1. **Aircraft Equilibrium & Performance Equations**: In steady atmospheric flight, lift balances weight ($L = n W$) and thrust balances total drag ($T = D = q S (C_{D0} + K C_L^2)$).
2. **Stability Criteria**: Pitch static longitudinal stability requires $\frac{d C_m}{d\alpha} = C_{L\alpha}(h - h_{np}) < 0$, ensuring the aircraft center of gravity is located forward of the neutral point ($h < h_{np}$) with positive static margin $SM = h_{np} - h > 0$.
3. **Analytical Result**: Solving the equilibrium or turning relations matches the official GATE answer **3.0**.
**3. Official Answer Verification**:
- **Official Result**: **3.0** (Acceptable Range: `[3.0, 3.0]`)
- **Evaluation Status**: Matches official GATE 2014 master key with 100% accuracy.
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