Question

Four free-vibration responses \(P,Q,R,S\) of a single-DOF spring-mass-damper with the same initial conditions are shown. Identify which curve corresponds to: (1) Overdamped, (2) Underdamped, (3) Critically damped, (4) Undamped. 

• A. \(P\!-\!1,\ Q\!-\!4,\ R\!-\!2,\ S\!-\!3\)
• B. \(P\!-\!1,\ Q\!-\!2,\ R\!-\!4,\ S\!-\!3\)
• C. \(P\!-\!3,\ Q\!-\!4,\ R\!-\!2,\ S\!-\!1\)
• D. \(P\!-\!3,\ Q\!-\!2,\ R\!-\!4,\ S\!-\!1\)

Hint:

Quick visual test: constant-amplitude sinusoid \(\Rightarrow\) undamped; oscillation with shrinking envelope \(\Rightarrow\) underdamped; among monotonic curves, the faster one is critical, the slower one is overdamped.

Answer 1

The Correct Option is A

Step 1: Qualitative signatures.

• Undamped (4): Pure sinusoid with constant amplitude (no decay). 
• Underdamped (2): Oscillatory with exponentially decaying amplitude. 
• Critically damped (3): No oscillation; fastest monotonic return to equilibrium. 
• Overdamped (1): No oscillation; slower monotonic return (two real distinct roots). 

Step 2: Read the figure against the signatures.

•  \(Q\) shows a sustained oscillation with essentially constant envelope \(\Rightarrow\) Undamped (4).
• Curve \(R\) shows oscillations whose amplitude diminishes with time \(\Rightarrow\) Underdamped (2). 
• Of the two non-oscillatory curves, \(S\) drops to equilibrium the quickest \(\Rightarrow\) Critically damped (3). 
• The remaining non-oscillatory curve \(P\) returns more slowly \(\Rightarrow\) Overdamped (1). 

 \[\boxed{P\!-\!1,\ Q\!-\!4,\ R\!-\!2,\ S\!-\!3}\]