Subtopic: Higher Order Linear ODEs with Constant Coefficients
Question Statement:
A general aviation airplane is initially in steady and level flight. The stability
coefficient
𝜕Cm
𝜕q (Cm is the airplane pitching moment coefficient and q is the pitch
rate) is negative. The airplane is perturbed with a small nose-up constant pitch
rate. Assume that the horizontal tail does not stall during the perturbation, and
unsteady effects are neglected. When compared to steady and level flight
condition, which of the following statements is/are true during the perturbed
motion?
Options:
(A) The angle of attack of the horizontal tail increases.
(B) The contribution of the horizontal tail to the airplane's pitching moment about the
center of gravity is more stabilizing.
(C) The lift generated by the horizontal tail increases.
(D) The contribution of the horizontal tail to the airplane's pitching moment about the
center of gravity is less stabilizing.
Official Answer Key:
A, B, C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2025 Q.38)**
**1. Intuitive Physical Setup & Given Parameters**:
- Airplane is initially in steady, level flight.
- Pitch rate damping derivative: $\frac{\partial C_m}{\partial q} < 0$.
- Aircraft is perturbed with a small, constant **nose-up pitch rate** ($q > 0$).
- Horizontal tail does not stall; unsteady effects neglected.
We need to determine the changes in the horizontal tail flow conditions and stability contributions during this perturbed motion (MSQ).
**2. Step-by-Step Flight Dynamics Derivation**:
1. **Kinematics of Nose-Up Pitch Rate at the Horizontal Tail**:
- The horizontal tail is located at distance $l_t$ behind the center of gravity ($l_t > 0$).
- A positive (nose-up) pitch rate $q$ causes the tail to move **downwards** with linear vertical velocity:
$w_{\text{tail}} = q l_t > 0 \quad (\text{downward relative to aircraft})$
- From the perspective of the horizontal tail airfoil, this downward motion induces an **upward relative airflow component** $\Delta w = q l_t$.
- The effective angle of attack of the horizontal tail increases by:
$\Delta \alpha_t = \frac{q l_t}{V} > 0$
- Thus, the angle of attack of the horizontal tail increases. Statement A is **TRUE**.
2. **Horizontal Tail Lift Contribution**:
- Because the tail operates below stall and $\Delta \alpha_t > 0$, the lift produced by the horizontal tail increases:
$\Delta L_t = q_\infty S_t C_{L\alpha, t} \Delta \alpha_t > 0$
- Statement C is **TRUE**.
3. **Horizontal Tail Pitching Moment Contribution**:
- An increase in upward lift on the aft-mounted horizontal tail generates a **nose-down (restoring) pitching moment** about the center of gravity:
$\Delta M_{\text{cg}} = -l_t \Delta L_t < 0$
- This restoring moment opposes the disturbing nose-up pitch rate $q$, providing pitch damping ($\partial C_m / \partial q < 0$).
- Therefore, the horizontal tail's contribution to pitching moment about the CG is **more stabilizing**.
- Statement B is **TRUE**, Statement D is **FALSE**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) The angle of attack of the horizontal tail increases**: ✅ **CORRECT**. Downward velocity of the tail induces an upward relative wind increment $\Delta \alpha_t = q l_t / V$.
- **(B) The contribution of the horizontal tail to the airplane's pitching moment about the center of gravity is more stabilizing**: ✅ **CORRECT**. Generates a restoring nose-down moment that opposes pitch rate.
- **(C) The lift generated by the horizontal tail increases**: ✅ **CORRECT**. $\Delta L_t = C_{L\alpha, t} \Delta \alpha_t > 0$.
- **(D) The contribution of the horizontal tail to the airplane's pitching moment about the center of gravity is less stabilizing**: ❌ **INCORRECT**. Directly contradicts Option B.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B, C**
- **Exam Takeaway**: Pitch damping ($C_{mq}$) arises from tail plunging kinematics: nose-up pitch pushes tail down, increasing tail angle of attack and lift, producing stabilizing nose-down pitch restoring moment.
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