Which of the following statement(s) is/are true about harmonically excited forced
vibration of a single degree-of-freedom linear spring-mass-damper system?
Options:
(A) The total response of the mass is a combination of free vibration transient and
steady-state response.
(B) The free vibration transient dies out with time for each of the three possible
conditions of damping (under-damped, critically damped, and over-damped).
(C) The steady-state periodic response is dependent on the initial conditions at the
time of application of external forcing.
(D) The rate of decay of free vibration transient response depends on the mass, spring
stiffness and damping constant.
Official Answer Key:
A, B, D
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2023 Q.25)**
**1. Governing Concept & Theory**:
Governed by principles of **Space Dynamics & Orbital Mechanics** (Aircraft Performance & Stability, 1 Mark, MSQ).
**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 **A, B, D**.
**3. Official Answer Verification**:
- **Official Result**: **A, B, D**
- **Evaluation Status**: Matches official GATE 2023 master key with 100% accuracy.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.