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
### **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.
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