GATE AE 2019 • Q54 NAT • 2 Marks

Flight Mechanics & Space Dynamics

Static Stability & Control

Subtopic: Directional Static Stability & Vertical Tail Volume Ratio

Question Statement:

The design of an airplane is modified to increase the vertical tail area by 20% and decrease the moment arm of the vertical tail by 20%. The ratio of the new directional stability derivative $C_{n\beta}$ (due to the vertical tail) to the original value is _________ (round off to 2 decimal places).

Official Answer Key:

0.96

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2019 Q.54 / AE Q.44)** **1. Intuitive Physical Setup & Given Parameters**: - Vertical tail contribution to directional stability derivative: $C_{n\beta, v} = \eta_v V_v C_{L\alpha, v} \left(1 + \frac{d\sigma}{d\beta}\right)$ - Vertical tail volume ratio: $V_v = \frac{S_v l_v}{S b}$ - Modification: - Vertical tail area increased by 20%: $S_v' = 1.20 S_v$ - Moment arm decreased by 20%: $l_v' = 0.80 l_v$ - Wing span $b$ and area $S$ remain unchanged. **2. Step-by-Step Mathematical Derivation**: 1. **Ratio of New to Old Vertical Tail Volume**: $V_v' = \frac{S_v' l_v'}{S b} = \frac{(1.20 S_v)(0.80 l_v)}{S b} = (1.20 \times 0.80) \left( \frac{S_v l_v}{S b} \right) = 0.96 V_v$ 2. **Directional Stability Derivative Ratio**: Since all other aerodynamic parameters remain constant: $\frac{C_{n\beta}'}{C_{n\beta}} = \frac{V_v'}{V_v} = 1.20 \times 0.80 = 0.96$ **3. Comprehensive Options & Distractor Analysis**: - **Question Type**: Numerical Answer Type (NAT). - **Exact Analytical Evaluation**: **0.96** (Acceptable Range: [0.96, 0.96]). **4. Official Master Key & Exam Takeaway**: - **Official Answer**: **0.96** - **Exam Rule**: $V_v \propto S_v \cdot l_v \implies (1+0.20)(1-0.20) = 1 - 0.04 = 0.96$.

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