Question

A spring–mass–damper system (mass $m$, stiffness $k$, damping coefficient $c$) excited by a force $F(t)=B\sin \omega t$ is shown in the figure. Four different responses of the system (marked as (i)–(iv)) are shown. In the figures, $A$ is the amplitude of the oscillatory response, and the dashed lines show its envelope. The responses represent only qualitative trends. Four different forcing and parameter conditions are also given:

\[ \text{(P)}\ c>0,\quad \omega=\sqrt{\frac{k}{m}}, \qquad \text{(Q)}\ c<0\ \text{and}\ \omega\ne 0, \] \[ \text{(R)}\ c=0,\quad \omega=\sqrt{\frac{k}{m}}, \qquad \text{(S)}\ c=0,\quad \omega \approx \sqrt{\frac{k}{m}}. \] Which option correctly matches each condition to its response (i)–(iv)?

• A. (P) $\to$ (i), (Q) $\to$ (iii), (R) $\to$ (iv), (S) $\to$ (ii)
• B. (P) $\to$ (ii), (Q) $\to$ (iii), (R) $\to$ (iv), (S) $\to$ (i)
• C. (P) $\to$ (i), (Q) $\to$ (iv), (R) $\to$ (ii), (S) $\to$ (iii)
• D. (P) $\to$ (iii), (Q) $\to$ (iv), (R) $\to$ (ii), (S) $\to$ (i)

Hint:

For forced vibrations: positive damping causes decay, zero damping at resonance gives constant amplitude, slight detuning creates beats, and negative damping leads to unbounded growth.

Answer 1

The Correct Option is C

Step 1: Identify response (i) — decaying oscillation.
Response (i) shows a gradually decreasing amplitude with time, indicating positive damping ($c>0$). Since the oscillations are still present and match the forcing frequency near resonance, this corresponds to \[ \text{(P)}\ c>0,\quad \omega=\sqrt{k/m}. \] So, (P) → (i). 
Step 2: Identify response (ii) — constant amplitude forced response (no damping). 
Response (ii) shows a steady oscillation with constant envelope. This occurs for: \[ c=0,\ \omega=\sqrt{k/m}, \] i.e., resonance in an undamped system (amplitude grows until limited by nonlinearity or steady-state representation). Thus, (R) → (ii). 
Step 3: Identify response (iii) — large amplitude beating pattern. 
Response (iii) shows large beats—a superposition pattern indicating two close frequencies (forcing and natural). This happens when \[ c=0,\ \omega \approx \sqrt{k/m}, \] but not exactly equal (slight detuning), giving beating. Thus, (S) → (iii). 
Step 4: Identify response (iv) — increasing amplitude. 
Response (iv) shows the amplitude growing unbounded with time. This happens when damping is negative ($c<0$), meaning the system supplies energy instead of removing it. Thus,

\[\text{(Q)}\ c<0\ \text{and}\ \omega\ne 0\quad \Rightarrow\quad (Q) → (iv).\]

Step 5: Final matching. 
\[ \boxed{\text{(P)→(i),\quad (Q)→(iv),\quad (R)→(ii),\quad (S)→(iii)}} \] which corresponds to option (C).