For a single degree of freedom spring-mass-damper system subjected to harmonic
forcing, the part of the motion (response) that decays due to damping is known as:
Options:
(A) transient response
(B) steady-state response
(C) harmonic response
(D) non-transient response
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.15)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Numerical Methods** (Newton-Raphson Method) within the *Engineering Mathematics* curriculum (1 Mark, MCQ).
**Problem Statement**:
> For a single degree of freedom spring-mass-damper system subjected to harmonic forcing, the part of the motion (response) that decays due to damping is known as:
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Engineering Mathematics. 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`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"transient response"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
- **(B)** *"steady-state response"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"harmonic response"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"non-transient response"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2024 (IISc Bengaluru) key.
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