Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2023 Q.25)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Space Dynamics & Orbital Mechanics** (Kepler's Laws & Two-Body Problem) within the *Flight Mechanics & Space Dynamics* curriculum (1 Mark, MSQ).
**Problem Statement**:
> 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?
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Flight Mechanics & Space Dynamics. Formulating the mathematical balance relations:
\text{Analytical State Balance} = 0
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`A, B, D`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"The total response of the mass is a combination of free vibration transient and
steady-state response."*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(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)."*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
- **(C)** *"The steady-state periodic response is dependent on the initial conditions at the
time of application of external forcing."*: ❌ **INCORRECT**. Contradicts governing physical laws.
- **(D)** *"The rate of decay of free vibration transient response depends on the mass, spring
stiffness and damping constant."*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws.
*Full credit (+1 Marks, 0 Negative Marks) requires selecting all verified choices **`A, B, D`**.*
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A, B, D**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2023 (IIT Kanpur) key.