(D) 14.52
Common Data Questions
Common Data for Questions 48 and 49:
The partial differential equation (PDE) governing free vibrations of a uniform Euler-Bernoulli beam is
given by
02
2
4
4
t
wmx
wEI , where EI is the flexural stiffness, is the mass per unit length,
is the bending displacement,
m
) , (t x w x is the coordinate along the beam length, t is time and is the beam
length.
L
To solve the PDE, the number of boundary
conditions (BC) and initial conditions (IC) needed are Q.48
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2011 Q.47)**
**1. Governing Concept & Theory**:
Governed by principles of **Atmospheric Flight Mechanics** (Aircraft Performance & Stability, 2 Mark, MCQ).
**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 **C**.
**3. Official Answer Verification**:
- **Official Result**: **C**
- **Evaluation Status**: Matches official GATE 2011 master key with 100% accuracy.
- **Selected Option**: **(C)** — *"7.26"*
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