A single degree of freedom spring-mass-damper system is designed to return to its equilibrium position in the shortest possible time without exhibiting oscillatory overshoot. The damping ratio $\zeta$ of this system must be:
Options:
(A) $\zeta = 0$
(B) $\zeta < 1$
(C) $\zeta = 1$
(D) $\zeta > 1$
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.19 / AE Q.9)**
**1. Intuitive Physical Setup & Given Parameters**:
- Single degree of freedom (SDOF) damped mechanical oscillator.
- Design criterion: Return to equilibrium in the shortest possible time without any oscillatory overshoot.
**2. Step-by-Step Vibration Dynamics Derivation**:
1. **Regimes of Damped SDOF Response**:
- **Undamped ($\zeta = 0$)**: Harmonic steady oscillation without decay.
- **Underdamped ($0 < \zeta < 1$)**: Decaying oscillatory motion that crosses the equilibrium position (exhibits overshoot).
- **Overdamped ($\zeta > 1$)**: Non-oscillatory exponential return, but sluggish due to high viscous drag (slow recovery).
- **Critically Damped ($\zeta = 1$)**: Boundary between oscillatory and non-oscillatory motion. Possesses repeated real characteristic roots:
$s_{1,2} = -\omega_n$
Solution: $x(t) = (C_1 + C_2 t)e^{-\omega_n t}$.
2. **Optimality of Critical Damping**:
- Critical damping achieves the fastest exponential decay rate without passing through zero or overshooting.
- Therefore, the required damping ratio is strictly **$\zeta = 1$**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** $\zeta = 0$: ❌ Incorrect. Undamped oscillation.
- **(B)** $\zeta < 1$: ❌ Incorrect. Oscillates and overshoots.
- **(C)** $\zeta = 1$: ✅ **CORRECT**. Definition of critical damping (fastest non-oscillatory return).
- **(D)** $\zeta > 1$: ❌ Incorrect. Sluggish, takes longer to settle than $\zeta = 1$.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **C**
- **Exam Rule**: Critical damping $\zeta = 1$ gives the fastest return to equilibrium without overshoot.
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